measure_theory.function.strongly_measurable.basic
⟷
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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We prove the "unfolding trick": Given a subgroup Γ
of a group G
, the integral of a function f
on G
times the lift to G
of a function g
on the coset space G ⧸ Γ
with respect to a right-invariant measure μ
on G
, is equal to the integral over the coset space of the automorphization of f
times g
.
We also prove the following simplified version: Given a subgroup Γ
of a group G
, the integral of a function f
on G
with respect to a right-invariant measure μ
is equal to the integral over the coset space G ⧸ Γ
of the automorphization of f
.
A question: is it possible to deduce ae_strongly_measurable (quotient_group.automorphize f) μ_𝓕
from ae_strongly_measurable f μ
(as opposed to assuming it as a hypothesis in the main theorem)? It seems quite plausible...
Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com>
Co-authored-by: Alex Kontorovich <58564076+AlexKontorovich@users.noreply.github.com> Co-authored-by: AlexKontorovich <58564076+AlexKontorovich@users.noreply.github.com>
@@ -1740,6 +1740,13 @@ end
end ae_strongly_measurable
+lemma ae_strongly_measurable_of_absolutely_continuous {α β : Type*} [measurable_space α]
+ [topological_space β] {μ ν : measure α} (h : ν ≪ μ) (g : α → β)
+ (hμ : ae_strongly_measurable g μ) : ae_strongly_measurable g ν :=
+begin
+ obtain ⟨g₁, hg₁, hg₁'⟩ := hμ,
+ refine ⟨g₁, hg₁, h.ae_eq hg₁'⟩,
+end
/-! ## Almost everywhere finitely strongly measurable functions -/
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is_unit c
from c ≠ 0
(#19081)
We already have this generalization for measurable_const_smul_iff
and ae_measurable_const_smul_iff
.
@@ -445,24 +445,24 @@ continuous_smul.comp_strongly_measurable (hf.prod_mk strongly_measurable_const)
end arithmetic
section mul_action
-
-variables [topological_space β] {G : Type*} [group G] [mul_action G β]
- [has_continuous_const_smul G β]
+variables {M G G₀ : Type*}
+variables [topological_space β]
+variables [monoid M] [mul_action M β] [has_continuous_const_smul M β]
+variables [group G] [mul_action G β] [has_continuous_const_smul G β]
+variables [group_with_zero G₀] [mul_action G₀ β] [has_continuous_const_smul G₀ β]
lemma _root_.strongly_measurable_const_smul_iff {m : measurable_space α} (c : G) :
strongly_measurable (λ x, c • f x) ↔ strongly_measurable f :=
⟨λ h, by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, λ h, h.const_smul c⟩
-variables {G₀ : Type*} [group_with_zero G₀] [mul_action G₀ β]
- [has_continuous_const_smul G₀ β]
+lemma _root_.is_unit.strongly_measurable_const_smul_iff {m : measurable_space α} {c : M}
+ (hc : is_unit c) :
+ strongly_measurable (λ x, c • f x) ↔ strongly_measurable f :=
+let ⟨u, hu⟩ := hc in hu ▸ strongly_measurable_const_smul_iff u
lemma _root_.strongly_measurable_const_smul_iff₀ {m : measurable_space α} {c : G₀} (hc : c ≠ 0) :
strongly_measurable (λ x, c • f x) ↔ strongly_measurable f :=
-begin
- refine ⟨λ h, _, λ h, h.const_smul c⟩,
- convert h.const_smul' c⁻¹,
- simp [smul_smul, inv_mul_cancel hc]
-end
+(is_unit.mk0 _ hc).strongly_measurable_const_smul_iff
end mul_action
@@ -1667,23 +1667,22 @@ end normed_space
section mul_action
-variables {G : Type*} [group G] [mul_action G β]
- [has_continuous_const_smul G β]
+variables {M G G₀ : Type*}
+variables [monoid M] [mul_action M β] [has_continuous_const_smul M β]
+variables [group G] [mul_action G β] [has_continuous_const_smul G β]
+variables [group_with_zero G₀] [mul_action G₀ β] [has_continuous_const_smul G₀ β]
lemma _root_.ae_strongly_measurable_const_smul_iff (c : G) :
ae_strongly_measurable (λ x, c • f x) μ ↔ ae_strongly_measurable f μ :=
⟨λ h, by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, λ h, h.const_smul c⟩
-variables {G₀ : Type*} [group_with_zero G₀] [mul_action G₀ β]
- [has_continuous_const_smul G₀ β]
+lemma _root_.is_unit.ae_strongly_measurable_const_smul_iff {c : M} (hc : is_unit c) :
+ ae_strongly_measurable (λ x, c • f x) μ ↔ ae_strongly_measurable f μ :=
+let ⟨u, hu⟩ := hc in hu ▸ ae_strongly_measurable_const_smul_iff u
lemma _root_.ae_strongly_measurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
ae_strongly_measurable (λ x, c • f x) μ ↔ ae_strongly_measurable f μ :=
-begin
- refine ⟨λ h, _, λ h, h.const_smul c⟩,
- convert h.const_smul' c⁻¹,
- simp [smul_smul, inv_mul_cancel hc]
-end
+(is_unit.mk0 _ hc).ae_strongly_measurable_const_smul_iff
end mul_action
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(first ported)
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -339,7 +339,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » t) -/
#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite /-
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
@@ -1046,9 +1046,9 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set /-
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
@@ -1074,7 +1074,7 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on /-
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -824,7 +824,7 @@ theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [Topologi
· let G : β → range g := cod_restrict g (range g) mem_range_self
have hG : ClosedEmbedding G :=
{ hg.cod_restrict _ _ with
- closed_range := by
+ isClosed_range := by
convert isClosed_univ
apply eq_univ_of_forall
rintro ⟨-, ⟨x, rfl⟩⟩
@@ -1954,7 +1954,7 @@ theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
· let G : β → range g := cod_restrict g (range g) mem_range_self
have hG : ClosedEmbedding G :=
{ hg.cod_restrict _ _ with
- closed_range := by
+ isClosed_range := by
convert isClosed_univ
apply eq_univ_of_forall
rintro ⟨-, ⟨x, rfl⟩⟩
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -254,7 +254,7 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
have h_tendsto := hf.tendsto_approx x
simp only [strongly_measurable.approx_bounded, simple_func.coe_map, Function.comp_apply]
by_cases hfx0 : ‖f x‖ = 0
- · rw [norm_eq_zero] at hfx0
+ · rw [norm_eq_zero] at hfx0
rw [hfx0] at h_tendsto ⊢
have h_tendsto_norm : tendsto (fun n => ‖hf.approx n x‖) at_top (𝓝 0) :=
by
@@ -328,7 +328,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
have : ∀ n, ∃ c, ∀ x, fs n x = c := fun n => simple_func.simple_func_bot (fs n)
let cs n := (this n).some
have h_cs_eq : ∀ n, ⇑(fs n) = fun x => cs n := fun n => funext (this n).choose_spec
- simp_rw [h_cs_eq] at h_fs_tendsto
+ simp_rw [h_cs_eq] at h_fs_tendsto
have h_tendsto : tendsto cs at_top (𝓝 (f hα.some)) := h_fs_tendsto hα.some
ext1 x
exact tendsto_nhds_unique (h_fs_tendsto x) h_tendsto
@@ -362,10 +362,10 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
refine' fun n => (measure_bUnion_finset_le _ _).trans_lt _
refine' ennreal.sum_lt_top_iff.mpr fun y hy => _
rw [simple_func.restrict_preimage_singleton _ ((hS_meas n).inter ht)]
- swap; · rw [Finset.mem_filter] at hy ; exact hy.2
+ swap; · rw [Finset.mem_filter] at hy; exact hy.2
refine' (measure_mono (Set.inter_subset_left _ _)).trans_lt _
have h_lt_top := measure_spanning_sets_lt_top (μ.restrict t) n
- rwa [measure.restrict_apply' ht] at h_lt_top
+ rwa [measure.restrict_apply' ht] at h_lt_top
· by_cases hxt : x ∈ t
swap; · rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]; exact tendsto_const_nhds
have h : tendsto (fun n => (f_approx n) x) at_top (𝓝 (f x)) := hf_meas.tendsto_approx x
@@ -642,7 +642,7 @@ variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : Mea
theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable l.Prod := by
induction' l with f l ihl; · exact strongly_measurable_one
- rw [List.forall_mem_cons] at hl
+ rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
@@ -781,7 +781,7 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
by
apply ClosedEmbedding.measurableEmbedding
exact closedEmbedding_subtype_val isClosed_closure
- have g_meas : Measurable g := by rw [fg] at H ; exact T.measurable_comp_iff.1 H
+ have g_meas : Measurable g := by rw [fg] at H; exact T.measurable_comp_iff.1 H
have : second_countable_topology (closure (range f)) :=
by
suffices separable_space (closure (range f)) by
@@ -852,7 +852,7 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
(is_separable_Union fun i => (hf (v i)).isSeparable_range).closure
apply this.mono
rintro _ ⟨x, rfl⟩
- rw [tendsto_pi_nhds] at lim
+ rw [tendsto_pi_nhds] at lim
apply mem_closure_of_tendsto ((limUnder x).comp hv)
apply eventually_of_forall fun n => _
apply mem_Union_of_mem n
@@ -1229,7 +1229,7 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
· have h_fs_zero : ∀ n, ∀ x ∈ tᶜ, fs n x = 0 :=
by
intro n x hxt
- rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
+ rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
simpa using hxt n
refine' fun x hxt => tendsto_nhds_unique (h_approx x) _
rw [funext fun n => h_fs_zero n x hxt]
@@ -1664,7 +1664,7 @@ theorem List.aestronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
by
induction' l with f l ihl; · exact ae_strongly_measurable_one
- rw [List.forall_mem_cons] at hl
+ rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
@@ -1919,7 +1919,7 @@ theorem aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace
refine' ⟨fun H => ⟨H.AEMeasurable, H.isSeparable_ae_range⟩, _⟩
rintro ⟨H, ⟨t, t_sep, ht⟩⟩
rcases eq_empty_or_nonempty t with (rfl | h₀)
- · simp only [mem_empty_iff_false, eventually_false_iff_eq_bot, ae_eq_bot] at ht
+ · simp only [mem_empty_iff_false, eventually_false_iff_eq_bot, ae_eq_bot] at ht
rw [ht]
exact ae_strongly_measurable_zero_measure f
· obtain ⟨g, g_meas, gt, fg⟩ : ∃ g : α → β, Measurable g ∧ range g ⊆ t ∧ f =ᵐ[μ] g :=
@@ -2014,7 +2014,7 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
have Hg : ae_strongly_measurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.strongly_measurable_mk, _⟩
filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x
- rwa [h'x] at hx
+ rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-/
@@ -2199,20 +2199,20 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
have A : MeasurableSet {x : α | f x ≠ 0} := (hf (measurable_set_singleton 0)).compl
refine' ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.strongly_measurable.smul g'meas, _⟩
apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ {x | f x ≠ 0}
- · rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
+ · rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
filter_upwards [hg'] with a ha h'a
have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, ENNReal.coe_eq_zero] using h'a
rw [ha this]
· filter_upwards [ae_restrict_mem A.compl] with x hx
- simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
+ simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
simp [hx]
· rintro ⟨g', g'meas, hg'⟩
refine' ⟨fun x => (f x : ℝ)⁻¹ • g' x, hf.coe_nnreal_real.inv.strongly_measurable.smul g'meas, _⟩
rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal]
filter_upwards [hg'] with x hx h'x
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
- simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
+ simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iff
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -1481,11 +1481,11 @@ theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν
#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure
-/
-#print MeasureTheory.AEStronglyMeasurable.mono' /-
-protected theorem mono' {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
+#print MeasureTheory.AEStronglyMeasurable.mono_ac /-
+protected theorem mono_ac {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
AEStronglyMeasurable f ν :=
⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'
+#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono_ac
-/
#print MeasureTheory.AEStronglyMeasurable.mono_set /-
@@ -2219,12 +2219,16 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
end AeStronglyMeasurable
-theorem aEStronglyMeasurable_of_absolutelyContinuous {α β : Type _} [MeasurableSpace α]
+/- warning: measure_theory.ae_strongly_measurable_of_absolutely_continuous clashes with measure_theory.ae_strongly_measurable.mono' -> MeasureTheory.AEStronglyMeasurable.mono_ac
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.AEStronglyMeasurable.mono_acₓ'. -/
+#print MeasureTheory.AEStronglyMeasurable.mono_ac /-
+theorem MeasureTheory.AEStronglyMeasurable.mono_ac {α β : Type _} [MeasurableSpace α]
[TopologicalSpace β] {μ ν : Measure α} (h : ν ≪ μ) (g : α → β) (hμ : AEStronglyMeasurable g μ) :
AEStronglyMeasurable g ν := by
obtain ⟨g₁, hg₁, hg₁'⟩ := hμ
refine' ⟨g₁, hg₁, h.ae_eq hg₁'⟩
-#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.aEStronglyMeasurable_of_absolutelyContinuous
+#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.AEStronglyMeasurable.mono_ac
+-/
/-! ## Almost everywhere finitely strongly measurable functions -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -169,11 +169,11 @@ theorem SimpleFunc.stronglyMeasurable {α β} {m : MeasurableSpace α} [Topologi
#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable
-/
-#print MeasureTheory.stronglyMeasurable_of_isEmpty /-
-theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {m : MeasurableSpace α} [TopologicalSpace β]
- (f : α → β) : StronglyMeasurable f :=
+#print MeasureTheory.StronglyMeasurable.of_finite /-
+theorem MeasureTheory.StronglyMeasurable.of_finite [IsEmpty α] {m : MeasurableSpace α}
+ [TopologicalSpace β] (f : α → β) : StronglyMeasurable f :=
⟨fun n => SimpleFunc.ofIsEmpty, isEmptyElim⟩
-#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
+#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.StronglyMeasurable.of_finite
-/
#print MeasureTheory.stronglyMeasurable_const /-
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -889,7 +889,48 @@ protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α →
theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
- (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by classical
+ (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by
+ classical
+ let f : ℕ → α →ₛ β := fun n =>
+ { toFun := fun x =>
+ if hx : x ∈ s then hc.approx n ⟨x, hx⟩
+ else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
+ measurableSet_fiber' := by
+ intro x
+ convert
+ (hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
+ ((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diffₓ hs)
+ ext1 y
+ simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
+ Subtype.coe_mk, exists_and_right, exists_eq_right, mem_diff]
+ by_cases hy : y ∈ s
+ · rw [dif_pos hy]
+ simp only [hy, exists_true_left, not_true, and_false_iff, or_false_iff]
+ · rw [dif_neg hy]
+ have A : y ∈ t := by simpa [hy] using h (mem_univ y)
+ simp only [A, hy, false_or_iff, IsEmpty.exists_iff, not_false_iff, and_true_iff,
+ exists_true_left]
+ finite_range' :=
+ by
+ apply ((hc.approx n).finite_range.union (hd.approx n).finite_range).Subset
+ rintro - ⟨y, rfl⟩
+ dsimp
+ by_cases hy : y ∈ s
+ · left
+ rw [dif_pos hy]
+ exact mem_range_self _
+ · right
+ rw [dif_neg hy]
+ exact mem_range_self _ }
+ refine' ⟨f, fun y => _⟩
+ by_cases hy : y ∈ s
+ · convert hc.tendsto_approx ⟨y, hy⟩ using 1
+ ext1 n
+ simp only [dif_pos hy, simple_func.apply_mk]
+ · have A : y ∈ t := by simpa [hy] using h (mem_univ y)
+ convert hd.tendsto_approx ⟨y, A⟩ using 1
+ ext1 n
+ simp only [dif_neg hy, simple_func.apply_mk]
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -889,48 +889,7 @@ protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α →
theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
- (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by
- classical
- let f : ℕ → α →ₛ β := fun n =>
- { toFun := fun x =>
- if hx : x ∈ s then hc.approx n ⟨x, hx⟩
- else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
- measurableSet_fiber' := by
- intro x
- convert
- (hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
- ((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diffₓ hs)
- ext1 y
- simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
- Subtype.coe_mk, exists_and_right, exists_eq_right, mem_diff]
- by_cases hy : y ∈ s
- · rw [dif_pos hy]
- simp only [hy, exists_true_left, not_true, and_false_iff, or_false_iff]
- · rw [dif_neg hy]
- have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- simp only [A, hy, false_or_iff, IsEmpty.exists_iff, not_false_iff, and_true_iff,
- exists_true_left]
- finite_range' :=
- by
- apply ((hc.approx n).finite_range.union (hd.approx n).finite_range).Subset
- rintro - ⟨y, rfl⟩
- dsimp
- by_cases hy : y ∈ s
- · left
- rw [dif_pos hy]
- exact mem_range_self _
- · right
- rw [dif_neg hy]
- exact mem_range_self _ }
- refine' ⟨f, fun y => _⟩
- by_cases hy : y ∈ s
- · convert hc.tendsto_approx ⟨y, hy⟩ using 1
- ext1 n
- simp only [dif_pos hy, simple_func.apply_mk]
- · have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- convert hd.tendsto_approx ⟨y, A⟩ using 1
- ext1 n
- simp only [dif_neg hy, simple_func.apply_mk]
+ (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by classical
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -614,7 +614,7 @@ open scoped Filter
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
- (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
-/
@@ -622,7 +622,7 @@ protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
- (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.inf MeasureTheory.StronglyMeasurable.inf
-/
@@ -1323,7 +1323,7 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
by
refine'
⟨fun n => hf.approx n ⊔ hg.approx n, fun n => _, fun x =>
- (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)⟩
refine' (measure_mono (support_sup _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
@@ -1335,7 +1335,7 @@ protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMe
by
refine'
⟨fun n => hf.approx n ⊓ hg.approx n, fun n => _, fun x =>
- (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)⟩
refine' (measure_mono (support_inf _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.inf
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,11 +3,11 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
-import Mathbin.Analysis.NormedSpace.FiniteDimension
-import Mathbin.Analysis.NormedSpace.BoundedLinearMaps
-import Mathbin.MeasureTheory.Constructions.BorelSpace.Metrizable
-import Mathbin.MeasureTheory.Integral.Lebesgue
-import Mathbin.MeasureTheory.Function.SimpleFuncDense
+import Analysis.NormedSpace.FiniteDimension
+import Analysis.NormedSpace.BoundedLinearMaps
+import MeasureTheory.Constructions.BorelSpace.Metrizable
+import MeasureTheory.Integral.Lebesgue
+import MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
@@ -339,7 +339,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » t) -/
#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite /-
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
@@ -1046,9 +1046,9 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set /-
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
@@ -1074,7 +1074,7 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on /-
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f
@@ -9,7 +9,7 @@ import Mathbin.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathbin.MeasureTheory.Integral.Lebesgue
import Mathbin.MeasureTheory.Function.SimpleFuncDense
-#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
+#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
/-!
# Strongly measurable and finitely strongly measurable functions
@@ -2219,6 +2219,13 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
end AeStronglyMeasurable
+theorem aEStronglyMeasurable_of_absolutelyContinuous {α β : Type _} [MeasurableSpace α]
+ [TopologicalSpace β] {μ ν : Measure α} (h : ν ≪ μ) (g : α → β) (hμ : AEStronglyMeasurable g μ) :
+ AEStronglyMeasurable g ν := by
+ obtain ⟨g₁, hg₁, hg₁'⟩ := hμ
+ refine' ⟨g₁, hg₁, h.ae_eq hg₁'⟩
+#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.aEStronglyMeasurable_of_absolutelyContinuous
+
/-! ## Almost everywhere finitely strongly measurable functions -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,11 +2,6 @@
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit ef95945cd48c932c9e034872bd25c3c220d9c946
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Analysis.NormedSpace.FiniteDimension
import Mathbin.Analysis.NormedSpace.BoundedLinearMaps
@@ -14,6 +9,8 @@ import Mathbin.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathbin.MeasureTheory.Integral.Lebesgue
import Mathbin.MeasureTheory.Function.SimpleFuncDense
+#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
+
/-!
# Strongly measurable and finitely strongly measurable functions
@@ -342,7 +339,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite /-
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
@@ -1049,9 +1046,9 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set /-
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
@@ -1077,7 +1074,7 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on /-
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -98,7 +98,6 @@ variable {α β γ ι : Type _} [Countable ι]
namespace MeasureTheory
--- mathport name: «expr →ₛ »
local infixr:25 " →ₛ " => SimpleFunc
section Definitions
@@ -112,7 +111,6 @@ def StronglyMeasurable [MeasurableSpace α] (f : α → β) : Prop :=
#align measure_theory.strongly_measurable MeasureTheory.StronglyMeasurable
-/
--- mathport name: strongly_measurable_of
scoped notation "strongly_measurable[" m "]" => @MeasureTheory.StronglyMeasurable _ _ _ m
#print MeasureTheory.FinStronglyMeasurable /-
@@ -146,12 +144,15 @@ open scoped MeasureTheory
/-! ## Strongly measurable functions -/
+#print MeasureTheory.StronglyMeasurable.aestronglyMeasurable /-
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {m0 : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
AEStronglyMeasurable f μ :=
⟨f, hf, EventuallyEq.refl _ _⟩
#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurable
+-/
+#print MeasureTheory.Subsingleton.stronglyMeasurable /-
@[simp]
theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton β] (f : α → β) : StronglyMeasurable f :=
@@ -162,29 +163,39 @@ theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [Topologica
rw [h_univ]
exact MeasurableSet.univ
#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable
+-/
+#print MeasureTheory.SimpleFunc.stronglyMeasurable /-
theorem SimpleFunc.stronglyMeasurable {α β} {m : MeasurableSpace α} [TopologicalSpace β]
(f : α →ₛ β) : StronglyMeasurable f :=
⟨fun _ => f, fun x => tendsto_const_nhds⟩
#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable
+-/
+#print MeasureTheory.stronglyMeasurable_of_isEmpty /-
theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {m : MeasurableSpace α} [TopologicalSpace β]
(f : α → β) : StronglyMeasurable f :=
⟨fun n => SimpleFunc.ofIsEmpty, isEmptyElim⟩
#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
+-/
+#print MeasureTheory.stronglyMeasurable_const /-
theorem stronglyMeasurable_const {α β} {m : MeasurableSpace α} [TopologicalSpace β] {b : β} :
StronglyMeasurable fun a : α => b :=
⟨fun n => SimpleFunc.const α b, fun a => tendsto_const_nhds⟩
#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_const
+-/
+#print MeasureTheory.stronglyMeasurable_one /-
@[to_additive]
theorem stronglyMeasurable_one {α β} {m : MeasurableSpace α} [TopologicalSpace β] [One β] :
StronglyMeasurable (1 : α → β) :=
@stronglyMeasurable_const _ _ _ _ 1
#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_one
#align measure_theory.strongly_measurable_zero MeasureTheory.stronglyMeasurable_zero
+-/
+#print MeasureTheory.stronglyMeasurable_const' /-
/-- A version of `strongly_measurable_const` that assumes `f x = f y` for all `x, y`.
This version works for functions between empty types. -/
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
@@ -194,12 +205,15 @@ theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalS
· exact strongly_measurable_of_is_empty f
· convert strongly_measurable_const; exact funext fun x => hf x h.some
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
+-/
+#print MeasureTheory.Subsingleton.stronglyMeasurable' /-
@[simp]
theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton α] (f : α → β) : StronglyMeasurable f :=
stronglyMeasurable_const' fun x y => by rw [Subsingleton.elim x y]
#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.stronglyMeasurable'
+-/
namespace StronglyMeasurable
@@ -218,10 +232,12 @@ protected noncomputable def approx {m : MeasurableSpace α} (hf : StronglyMeasur
#align measure_theory.strongly_measurable.approx MeasureTheory.StronglyMeasurable.approx
-/
+#print MeasureTheory.StronglyMeasurable.tendsto_approx /-
protected theorem tendsto_approx {m : MeasurableSpace α} (hf : StronglyMeasurable f) :
∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
hf.choose_spec
#align measure_theory.strongly_measurable.tendsto_approx MeasureTheory.StronglyMeasurable.tendsto_approx
+-/
#print MeasureTheory.StronglyMeasurable.approxBounded /-
/-- Similar to `strongly_measurable.approx`, but enforces that the norm of every function in the
@@ -233,6 +249,7 @@ noncomputable def approxBounded {m : MeasurableSpace α} [Norm β] [SMul ℝ β]
#align measure_theory.strongly_measurable.approx_bounded MeasureTheory.StronglyMeasurable.approxBounded
-/
+#print MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le /-
theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} (hf : strongly_measurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) :=
@@ -268,14 +285,18 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
refine' tendsto.min tendsto_const_nhds _
refine' tendsto.div tendsto_const_nhds h_tendsto.norm hfx0
#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
+-/
+#print MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae /-
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : strongly_measurable[m] f) {c : ℝ}
(hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
∀ᵐ x ∂μ, Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) := by
filter_upwards [hf_bound] with x hfx using tendsto_approx_bounded_of_norm_le hf hfx
#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
+-/
+#print MeasureTheory.StronglyMeasurable.norm_approxBounded_le /-
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : strongly_measurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
‖hf.approxBounded c n x‖ ≤ c :=
@@ -295,7 +316,9 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
inv_mul_cancel h0, one_mul, Real.norm_of_nonneg hc]
· rwa [div_le_one (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]
#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_le
+-/
+#print stronglyMeasurable_bot_iff /-
theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
strongly_measurable[⊥] f ↔ ∃ c, f = fun _ => c :=
by
@@ -315,10 +338,12 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
· obtain ⟨c, rfl⟩ := hf_eq
exact strongly_measurable_const
#align strongly_measurable_bot_iff stronglyMeasurable_bot_iff
+-/
end BasicPropertiesInAnyTopologicalSpace
/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » t) -/
+#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite /-
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
(ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
@@ -367,6 +392,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
rw [hn₁ m ((le_max_left _ _).trans hm.le)]
exact hn₂ m ((le_max_right _ _).trans hm.le)
#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
+-/
#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable /-
/-- If the measure is sigma-finite, all strongly measurable functions are
@@ -378,33 +404,42 @@ protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : Me
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
-/
+#print MeasureTheory.StronglyMeasurable.measurable /-
/-- A strongly measurable function is measurable. -/
protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f :=
measurable_of_tendsto_metrizable (fun n => (hf.approx n).Measurable)
(tendsto_pi_nhds.mpr hf.tendsto_approx)
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
+-/
+#print MeasureTheory.StronglyMeasurable.aemeasurable /-
/-- A strongly measurable function is almost everywhere measurable. -/
protected theorem aemeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
(hf : StronglyMeasurable f) : AEMeasurable f μ :=
hf.Measurable.AEMeasurable
#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurable
+-/
+#print Continuous.comp_stronglyMeasurable /-
theorem Continuous.comp_stronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => g (f x) :=
⟨fun n => SimpleFunc.map g (hf.approx n), fun x => (hg.Tendsto _).comp (hf.tendsto_approx x)⟩
#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurable
+-/
+#print MeasureTheory.StronglyMeasurable.measurableSet_mulSupport /-
@[to_additive]
theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
[MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) := by borelize β;
exact measurableSet_mulSupport hf.measurable
#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
#align measure_theory.strongly_measurable.measurable_set_support MeasureTheory.StronglyMeasurable.measurableSet_support
+-/
+#print MeasureTheory.StronglyMeasurable.mono /-
protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
(hf : strongly_measurable[m'] f) (h_mono : m' ≤ m) : strongly_measurable[m] f :=
by
@@ -414,7 +449,9 @@ protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
finite_range' := simple_func.finite_range (hf.approx n) }
exact ⟨f_approx, hf.tendsto_approx⟩
#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.mono
+-/
+#print MeasureTheory.StronglyMeasurable.prod_mk /-
protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ]
{f : α → β} {g : α → γ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => (f x, g x) :=
@@ -423,30 +460,35 @@ protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [Topolo
rw [nhds_prod_eq]
exact tendsto.prod_mk (hf.tendsto_approx x) (hg.tendsto_approx x)
#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mk
+-/
+#print MeasureTheory.StronglyMeasurable.comp_measurable /-
theorem comp_measurable [TopologicalSpace β] {m : MeasurableSpace α} {m' : MeasurableSpace γ}
{f : α → β} {g : γ → α} (hf : StronglyMeasurable f) (hg : Measurable g) :
StronglyMeasurable (f ∘ g) :=
⟨fun n => SimpleFunc.comp (hf.approx n) g hg, fun x => hf.tendsto_approx (g x)⟩
#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurable
+-/
+#print MeasureTheory.StronglyMeasurable.of_uncurry_left /-
theorem of_uncurry_left [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) :=
hf.comp_measurable measurable_prod_mk_left
#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_left
+-/
+#print MeasureTheory.StronglyMeasurable.of_uncurry_right /-
theorem of_uncurry_right [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
StronglyMeasurable fun x => f x y :=
hf.comp_measurable measurable_prod_mk_right
#align measure_theory.strongly_measurable.of_uncurry_right MeasureTheory.StronglyMeasurable.of_uncurry_right
+-/
section Arithmetic
variable {mα : MeasurableSpace α} [TopologicalSpace β]
-include mα
-
#print MeasureTheory.StronglyMeasurable.mul /-
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
@@ -474,12 +516,14 @@ theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c :
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
-/
+#print MeasureTheory.StronglyMeasurable.inv /-
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
⟨fun n => (hf.approx n)⁻¹, fun x => (hf.tendsto_approx x).inv⟩
#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv
#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg
+-/
#print MeasureTheory.StronglyMeasurable.div /-
@[to_additive]
@@ -537,21 +581,27 @@ variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+#print stronglyMeasurable_const_smul_iff /-
theorem stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iff
+-/
+#print IsUnit.stronglyMeasurable_const_smul_iff /-
theorem IsUnit.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} {c : M} (hc : IsUnit c) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
let ⟨u, hu⟩ := hc
hu ▸ stronglyMeasurable_const_smul_iff u
#align is_unit.strongly_measurable_const_smul_iff IsUnit.stronglyMeasurable_const_smul_iff
+-/
+#print stronglyMeasurable_const_smul_iff₀ /-
theorem stronglyMeasurable_const_smul_iff₀ {m : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
(IsUnit.mk0 _ hc).stronglyMeasurable_const_smul_iff
#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀
+-/
end MulAction
@@ -590,8 +640,7 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
-include m
-
+#print List.stronglyMeasurable_prod' /-
@[to_additive]
theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable l.Prod := by
@@ -601,13 +650,16 @@ theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, S
exact hl.1.mul (ihl hl.2)
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
#align list.strongly_measurable_sum' List.stronglyMeasurable_sum'
+-/
+#print List.stronglyMeasurable_prod /-
@[to_additive]
theorem List.stronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable fun x => (l.map fun f : α → M => f x).Prod := by
simpa only [← Pi.list_prod_apply] using l.strongly_measurable_prod' hl
#align list.strongly_measurable_prod List.stronglyMeasurable_prod
#align list.strongly_measurable_sum List.stronglyMeasurable_sum
+-/
end Monoid
@@ -615,15 +667,16 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
-include m
-
+#print Multiset.stronglyMeasurable_prod' /-
@[to_additive]
theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.Prod := by rcases l with ⟨l⟩;
simpa using l.strongly_measurable_prod' (by simpa using hl)
#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'
#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'
+-/
+#print Multiset.stronglyMeasurable_prod /-
@[to_additive]
theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, StronglyMeasurable f) :
@@ -631,23 +684,29 @@ theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
simpa only [← Pi.multiset_prod_apply] using s.strongly_measurable_prod' hs
#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prod
#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum
+-/
+#print Finset.stronglyMeasurable_prod' /-
@[to_additive]
theorem Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i in s, f i) :=
Finset.prod_induction _ _ (fun a b ha hb => ha.mul hb) (@stronglyMeasurable_one α M _ _ _) hf
#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'
#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'
+-/
+#print Finset.stronglyMeasurable_prod /-
@[to_additive]
theorem Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i in s, f i a := by
simpa only [← Finset.prod_apply] using s.strongly_measurable_prod' hf
#align finset.strongly_measurable_prod Finset.stronglyMeasurable_prod
#align finset.strongly_measurable_sum Finset.stronglyMeasurable_sum
+-/
end CommMonoid
+#print MeasureTheory.StronglyMeasurable.isSeparable_range /-
/-- The range of a strongly measurable function is separable. -/
theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : StronglyMeasurable f) :
TopologicalSpace.IsSeparable (range f) :=
@@ -661,20 +720,21 @@ theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : S
apply mem_Union_of_mem n
exact mem_range_self _
#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_range
+-/
+#print MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton /-
theorem separableSpace_range_union_singleton {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (hf : StronglyMeasurable f) {b : β} :
SeparableSpace (range f ∪ {b} : Set β) :=
letI := pseudo_metrizable_space_pseudo_metric β
(hf.is_separable_range.union (finite_singleton _).IsSeparable).SeparableSpace
#align measure_theory.strongly_measurable.separable_space_range_union_singleton MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton
+-/
section SecondCountableStronglyMeasurable
variable {mα : MeasurableSpace α} [MeasurableSpace β]
-include mα
-
#print Measurable.stronglyMeasurable /-
/-- In a space with second countable topology, measurable implies strongly measurable. -/
theorem Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
@@ -708,6 +768,7 @@ theorem stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
end SecondCountableStronglyMeasurable
+#print stronglyMeasurable_iff_measurable_separable /-
/-- A function is strongly measurable if and only if it is measurable and has separable
range. -/
theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [TopologicalSpace β]
@@ -733,7 +794,9 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
rw [fg]
exact continuous_subtype_coe.comp_strongly_measurable g_smeas
#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separable
+-/
+#print Continuous.stronglyMeasurable /-
/-- A continuous function is strongly measurable when either the source space or the target space
is second-countable. -/
theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
@@ -748,7 +811,9 @@ theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
exact (is_separable_of_separable_space univ).image hf
· exact hf.measurable.strongly_measurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
+-/
+#print Embedding.comp_stronglyMeasurable_iff /-
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
@@ -774,7 +839,9 @@ theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [Topologi
ext x
simp [hg.inj.eq_iff]
#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff
+-/
+#print stronglyMeasurable_of_tendsto /-
/-- A sequential limit of strongly measurable functions is strongly measurable. -/
theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β}
@@ -794,7 +861,9 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
apply mem_Union_of_mem n
exact mem_range_self _
#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendsto
+-/
+#print MeasureTheory.StronglyMeasurable.piecewise /-
protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
{_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) :=
@@ -804,7 +873,9 @@ protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s :
· simpa [hx] using hf.tendsto_approx x
· simpa [hx] using hg.tendsto_approx x
#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewise
+-/
+#print MeasureTheory.StronglyMeasurable.ite /-
/-- this is slightly different from `strongly_measurable.piecewise`. It can be used to show
`strongly_measurable (ite (x=0) 0 1)` by
`exact strongly_measurable.ite (measurable_set_singleton 0) strongly_measurable_const
@@ -815,7 +886,9 @@ protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α →
(hg : StronglyMeasurable g) : StronglyMeasurable fun x => ite (p x) (f x) (g x) :=
StronglyMeasurable.piecewise hp hf hg
#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite
+-/
+#print stronglyMeasurable_of_stronglyMeasurable_union_cover /-
theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
@@ -862,7 +935,9 @@ theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpac
ext1 n
simp only [dif_neg hy, simple_func.apply_mk]
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
+-/
+#print stronglyMeasurable_of_restrict_of_restrict_compl /-
theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
(h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
@@ -870,39 +945,53 @@ theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α
stronglyMeasurable_of_stronglyMeasurable_union_cover s (sᶜ) hs hs.compl (union_compl_self s).ge h₁
h₂
#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl
+-/
+#print MeasureTheory.StronglyMeasurable.indicator /-
protected theorem indicator {m : MeasurableSpace α} [TopologicalSpace β] [Zero β]
(hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
StronglyMeasurable (s.indicator f) :=
hf.piecewise hs stronglyMeasurable_const
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
+-/
+#print MeasureTheory.StronglyMeasurable.dist /-
protected theorem dist {m : MeasurableSpace α} {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist
+-/
+#print MeasureTheory.StronglyMeasurable.norm /-
protected theorem norm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ :=
continuous_norm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm
+-/
+#print MeasureTheory.StronglyMeasurable.nnnorm /-
protected theorem nnnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ :=
continuous_nnnorm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm
+-/
+#print MeasureTheory.StronglyMeasurable.ennnorm /-
protected theorem ennnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
(ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).Measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
+-/
+#print MeasureTheory.StronglyMeasurable.real_toNNReal /-
protected theorem real_toNNReal {m : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => (f x).toNNReal :=
continuous_real_toNNReal.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal
+-/
+#print MeasurableEmbedding.stronglyMeasurable_extend /-
theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
@@ -918,7 +1007,9 @@ theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α
simpa only [hx, simple_func.extend_apply', not_false_iff, extend_apply'] using
hg'.tendsto_approx x
#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
+-/
+#print MeasurableEmbedding.exists_stronglyMeasurable_extend /-
theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
@@ -927,14 +1018,18 @@ theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g
hg.stronglyMeasurable_extend hf (stronglyMeasurable_const' fun _ _ => rfl),
funext fun x => hg.Injective.extend_apply _ _ _⟩
#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend
+-/
+#print MeasureTheory.StronglyMeasurable.measurableSet_eq_fun /-
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
{f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
MeasurableSet {x | f x = g x} := by
borelize (E × E)
exact (hf.prod_mk hg).Measurable is_closed_diagonal.measurable_set
#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
+-/
+#print MeasureTheory.StronglyMeasurable.measurableSet_lt /-
theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet {a | f a < g a} :=
@@ -942,7 +1037,9 @@ theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearO
borelize (β × β)
exact (hf.prod_mk hg).Measurable is_open_lt_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_lt
+-/
+#print MeasureTheory.StronglyMeasurable.measurableSet_le /-
theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet {a | f a ≤ g a} :=
@@ -950,10 +1047,12 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
borelize (β × β)
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
+-/
/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set /-
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hf_zero : ∀ (x) (_ : x ∉ s), f x = 0) :
@@ -976,8 +1075,10 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
· simp_rw [hg_zero x hx, hf_zero x hx]
exact tendsto_const_nhds
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
+-/
/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+#print MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on /-
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
on `s` is `m`-strongly-measurable, then `f` is also `m₂`-strongly-measurable. -/
@@ -1018,7 +1119,9 @@ theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableS
simp_rw [hg_eq]
exact hg_seq_tendsto x
#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
+-/
+#print MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le /-
/-- If a function `f` is strongly measurable w.r.t. a sub-σ-algebra `m` and the measure is σ-finite
on `m`, then there exists spanning measurable sets with finite measure on which `f` has bounded
norm. In particular, `f` is integrable on each of those sets. -/
@@ -1054,12 +1157,14 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
exact_mod_cast hij
rw [this, norm_sets_spanning, Union_spanning_sets (μ.trim hm), Set.inter_univ]
#align measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le
+-/
end StronglyMeasurable
/-! ## Finitely strongly measurable functions -/
+#print MeasureTheory.finStronglyMeasurable_zero /-
theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
[TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ :=
⟨0, by
@@ -1067,6 +1172,7 @@ theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measur
WithTop.zero_lt_top, forall_const],
fun n => tendsto_const_nhds⟩
#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zero
+-/
namespace FinStronglyMeasurable
@@ -1092,9 +1198,11 @@ protected noncomputable def approx : ℕ → α →ₛ β :=
#align measure_theory.fin_strongly_measurable.approx MeasureTheory.FinStronglyMeasurable.approx
-/
+#print MeasureTheory.FinStronglyMeasurable.fin_support_approx /-
protected theorem fin_support_approx : ∀ n, μ (support (hf.approx n)) < ∞ :=
hf.choose_spec.1
#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approx
+-/
#print MeasureTheory.FinStronglyMeasurable.tendsto_approx /-
protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
@@ -1111,6 +1219,7 @@ protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
#align measure_theory.fin_strongly_measurable.strongly_measurable MeasureTheory.FinStronglyMeasurable.stronglyMeasurable
-/
+#print MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite /-
theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
(hf : FinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) :=
@@ -1135,6 +1244,7 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
· rw [← Set.union_iUnion (tᶜ) T]
exact Set.compl_union_self _
#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
+-/
#print MeasureTheory.FinStronglyMeasurable.measurable /-
/-- A finitely strongly measurable function is measurable. -/
@@ -1148,6 +1258,7 @@ section Arithmetic
variable [TopologicalSpace β]
+#print MeasureTheory.FinStronglyMeasurable.mul /-
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ :=
by
@@ -1157,7 +1268,9 @@ protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMe
intro n
exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul
+-/
+#print MeasureTheory.FinStronglyMeasurable.add /-
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ :=
⟨fun n => hf.approx n + hg.approx n, fun n =>
@@ -1166,7 +1279,9 @@ protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasura
(ENNReal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add
+-/
+#print MeasureTheory.FinStronglyMeasurable.neg /-
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
FinStronglyMeasurable (-f) μ :=
by
@@ -1175,7 +1290,9 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMe
rw [Function.support_neg (hf.approx n)]
exact hf.fin_support_approx n
#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg
+-/
+#print MeasureTheory.FinStronglyMeasurable.sub /-
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ :=
⟨fun n => hf.approx n - hg.approx n, fun n =>
@@ -1184,7 +1301,9 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
(ENNReal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
+-/
+#print MeasureTheory.FinStronglyMeasurable.const_smul /-
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
FinStronglyMeasurable (c • f) μ :=
@@ -1193,6 +1312,7 @@ protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Mono
rw [simple_func.coe_smul]
refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smul
+-/
end Arithmetic
@@ -1200,6 +1320,7 @@ section Order
variable [TopologicalSpace β] [Zero β]
+#print MeasureTheory.FinStronglyMeasurable.sup /-
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ :=
by
@@ -1209,7 +1330,9 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
refine' (measure_mono (support_sup _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
+-/
+#print MeasureTheory.FinStronglyMeasurable.inf /-
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ :=
by
@@ -1219,11 +1342,13 @@ protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMe
refine' (measure_mono (support_inf _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.inf
+-/
end Order
end FinStronglyMeasurable
+#print MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite /-
theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β}
[TopologicalSpace β] [T2Space β] [Zero β] {m : MeasurableSpace α} {μ : Measure α} :
FinStronglyMeasurable f μ ↔
@@ -1233,39 +1358,51 @@ theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
hf.1.finStronglyMeasurable_of_set_sigmaFinite hf.2.choose_spec.1 hf.2.choose_spec.2.1
hf.2.choose_spec.2.2⟩
#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
+-/
+#print MeasureTheory.aefinStronglyMeasurable_zero /-
theorem aefinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
[TopologicalSpace β] : AEFinStronglyMeasurable (0 : α → β) μ :=
⟨0, finStronglyMeasurable_zero, EventuallyEq.rfl⟩
#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aefinStronglyMeasurable_zero
+-/
/-! ## Almost everywhere strongly measurable functions -/
+#print MeasureTheory.aestronglyMeasurable_const /-
theorem aestronglyMeasurable_const {α β} {m : MeasurableSpace α} {μ : Measure α}
[TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun a : α => b) μ :=
stronglyMeasurable_const.AEStronglyMeasurable
#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_const
+-/
+#print MeasureTheory.aestronglyMeasurable_one /-
@[to_additive]
theorem aestronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
[One β] : AEStronglyMeasurable (1 : α → β) μ :=
stronglyMeasurable_one.AEStronglyMeasurable
#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_one
#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aestronglyMeasurable_zero
+-/
+#print MeasureTheory.Subsingleton.aestronglyMeasurable /-
@[simp]
theorem Subsingleton.aestronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[Subsingleton β] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
(Subsingleton.stronglyMeasurable f).AEStronglyMeasurable
#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurable
+-/
+#print MeasureTheory.Subsingleton.aestronglyMeasurable' /-
@[simp]
theorem Subsingleton.aestronglyMeasurable' {m : MeasurableSpace α} [TopologicalSpace β]
[Subsingleton α] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
(Subsingleton.stronglyMeasurable' f).AEStronglyMeasurable
#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'
+-/
+#print MeasureTheory.aestronglyMeasurable_zero_measure /-
@[simp]
theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
AEStronglyMeasurable f (0 : Measure α) :=
@@ -1274,11 +1411,14 @@ theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace
inhabit α
exact ⟨fun x => f default, strongly_measurable_const, rfl⟩
#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measure
+-/
+#print MeasureTheory.SimpleFunc.aestronglyMeasurable /-
theorem SimpleFunc.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
(f : α →ₛ β) : AEStronglyMeasurable f μ :=
f.StronglyMeasurable.AEStronglyMeasurable
#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aestronglyMeasurable
+-/
namespace AeStronglyMeasurable
@@ -1295,9 +1435,11 @@ protected noncomputable def mk (f : α → β) (hf : AEStronglyMeasurable f μ)
#align measure_theory.ae_strongly_measurable.mk MeasureTheory.AEStronglyMeasurable.mk
-/
+#print MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk /-
theorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) :=
hf.choose_spec.1
#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk
+-/
#print MeasureTheory.AEStronglyMeasurable.measurable_mk /-
theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
@@ -1306,9 +1448,11 @@ theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpac
#align measure_theory.ae_strongly_measurable.measurable_mk MeasureTheory.AEStronglyMeasurable.measurable_mk
-/
+#print MeasureTheory.AEStronglyMeasurable.ae_eq_mk /-
theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk
+-/
#print MeasureTheory.AEStronglyMeasurable.aemeasurable /-
protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
@@ -1320,47 +1464,64 @@ protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
end Mk
+#print MeasureTheory.AEStronglyMeasurable.congr /-
theorem congr (hf : AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AEStronglyMeasurable g μ :=
⟨hf.mk f, hf.stronglyMeasurable_mk, h.symm.trans hf.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congr
+-/
+#print aestronglyMeasurable_congr /-
theorem aestronglyMeasurable_congr (h : f =ᵐ[μ] g) :
AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g μ :=
⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩
#align ae_strongly_measurable_congr aestronglyMeasurable_congr
+-/
+#print MeasureTheory.AEStronglyMeasurable.mono_measure /-
theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν ≤ μ) :
AEStronglyMeasurable f ν :=
⟨hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure
+-/
+#print MeasureTheory.AEStronglyMeasurable.mono' /-
protected theorem mono' {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
AEStronglyMeasurable f ν :=
⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'
+-/
+#print MeasureTheory.AEStronglyMeasurable.mono_set /-
theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
AEStronglyMeasurable f (μ.restrict s) :=
ht.mono_measure (restrict_mono h le_rfl)
#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_set
+-/
+#print MeasureTheory.AEStronglyMeasurable.restrict /-
protected theorem restrict (hfm : AEStronglyMeasurable f μ) {s} :
AEStronglyMeasurable f (μ.restrict s) :=
hfm.mono_measure Measure.restrict_le_self
#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrict
+-/
+#print MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk /-
theorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (μ.restrict s)) :
∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x :=
ae_imp_of_ae_restrict h.ae_eq_mk
#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk
+-/
+#print Continuous.comp_aestronglyMeasurable /-
/-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable. -/
theorem Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => g (f x)) μ :=
⟨_, hg.comp_stronglyMeasurable hf.stronglyMeasurable_mk, EventuallyEq.fun_comp hf.ae_eq_mk g⟩
#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurable
+-/
+#print Continuous.aestronglyMeasurable /-
/-- A continuous function from `α` to `β` is ae strongly measurable when one of the two spaces is
second countable. -/
theorem Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
@@ -1368,19 +1529,24 @@ theorem Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSp
AEStronglyMeasurable f μ :=
hf.StronglyMeasurable.AEStronglyMeasurable
#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurable
+-/
+#print MeasureTheory.AEStronglyMeasurable.prod_mk /-
protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (fun x => (f x, g x)) μ :=
⟨fun x => (hf.mk f x, hg.mk g x), hf.stronglyMeasurable_mk.prod_mk hg.stronglyMeasurable_mk,
hf.ae_eq_mk.prod_mk hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk
+-/
+#print Measurable.aestronglyMeasurable /-
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
theorem Measurable.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [MeasurableSpace β]
[PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β]
(hf : Measurable f) : AEStronglyMeasurable f μ :=
hf.StronglyMeasurable.AEStronglyMeasurable
#align measurable.ae_strongly_measurable Measurable.aestronglyMeasurable
+-/
section Arithmetic
@@ -1412,13 +1578,16 @@ protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurab
#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AEStronglyMeasurable.const_add
-/
+#print MeasureTheory.AEStronglyMeasurable.inv /-
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable f⁻¹ μ :=
⟨(hf.mk f)⁻¹, hf.stronglyMeasurable_mk.inv, hf.ae_eq_mk.inv⟩
#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AEStronglyMeasurable.neg
+-/
+#print MeasureTheory.AEStronglyMeasurable.div /-
@[to_additive]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ :=
@@ -1426,6 +1595,7 @@ protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurabl
hf.ae_eq_mk.div hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.div
#align measure_theory.ae_strongly_measurable.sub MeasureTheory.AEStronglyMeasurable.sub
+-/
#print MeasureTheory.AEStronglyMeasurable.smul /-
@[to_additive]
@@ -1464,17 +1634,21 @@ end Arithmetic
section Order
+#print MeasureTheory.AEStronglyMeasurable.sup /-
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.stronglyMeasurable_mk.sup hg.stronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.sup
+-/
+#print MeasureTheory.AEStronglyMeasurable.inf /-
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.stronglyMeasurable_mk.inf hg.stronglyMeasurable_mk,
hf.ae_eq_mk.inf hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.inf MeasureTheory.AEStronglyMeasurable.inf
+-/
end Order
@@ -1487,6 +1661,7 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
+#print List.aestronglyMeasurable_prod' /-
@[to_additive]
theorem List.aestronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
@@ -1497,13 +1672,16 @@ theorem List.aestronglyMeasurable_prod' (l : List (α → M))
exact hl.1.mul (ihl hl.2)
#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
#align list.ae_strongly_measurable_sum' List.aestronglyMeasurable_sum'
+-/
+#print List.aestronglyMeasurable_prod /-
@[to_additive]
theorem List.aestronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).Prod) μ := by
simpa only [← Pi.list_prod_apply] using l.ae_strongly_measurable_prod' hl
#align list.ae_strongly_measurable_prod List.aestronglyMeasurable_prod
#align list.ae_strongly_measurable_sum List.aestronglyMeasurable_sum
+-/
end Monoid
@@ -1511,13 +1689,16 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
+#print Multiset.aestronglyMeasurable_prod' /-
@[to_additive]
theorem Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ := by
rcases l with ⟨l⟩; simpa using l.ae_strongly_measurable_prod' (by simpa using hl)
#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'
#align multiset.ae_strongly_measurable_sum' Multiset.aestronglyMeasurable_sum'
+-/
+#print Multiset.aestronglyMeasurable_prod /-
@[to_additive]
theorem Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
@@ -1525,7 +1706,9 @@ theorem Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
simpa only [← Pi.multiset_prod_apply] using s.ae_strongly_measurable_prod' hs
#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prod
#align multiset.ae_strongly_measurable_sum Multiset.aestronglyMeasurable_sum
+-/
+#print Finset.aestronglyMeasurable_prod' /-
@[to_additive]
theorem Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i in s, f i) μ :=
@@ -1534,7 +1717,9 @@ theorem Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s
hg ▸ hf _ hi
#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'
#align finset.ae_strongly_measurable_sum' Finset.aestronglyMeasurable_sum'
+-/
+#print Finset.aestronglyMeasurable_prod /-
@[to_additive]
theorem Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
@@ -1542,6 +1727,7 @@ theorem Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s
simpa only [← Finset.prod_apply] using s.ae_strongly_measurable_prod' hf
#align finset.ae_strongly_measurable_prod Finset.aestronglyMeasurable_prod
#align finset.ae_strongly_measurable_sum Finset.aestronglyMeasurable_sum
+-/
end CommMonoid
@@ -1590,10 +1776,12 @@ protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AEStronglyMeasurable.norm
-/
+#print MeasureTheory.AEStronglyMeasurable.nnnorm /-
protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ :=
continuous_nnnorm.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnorm
+-/
#print MeasureTheory.AEStronglyMeasurable.ennnorm /-
protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
@@ -1610,10 +1798,12 @@ protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α →
#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AEStronglyMeasurable.edist
-/
+#print MeasureTheory.AEStronglyMeasurable.real_toNNReal /-
protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (f x).toNNReal) μ :=
continuous_real_toNNReal.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AEStronglyMeasurable.real_toNNReal
+-/
#print aestronglyMeasurable_indicator_iff /-
theorem aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
@@ -1687,25 +1877,32 @@ theorem aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : Measurabl
#align ae_strongly_measurable_of_ae_strongly_measurable_trim aestronglyMeasurable_of_aestronglyMeasurable_trim
-/
+#print MeasureTheory.AEStronglyMeasurable.comp_aemeasurable /-
theorem comp_aemeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
AEStronglyMeasurable (g ∘ f) μ :=
⟨hg.mk g ∘ hf.mk f, hg.stronglyMeasurable_mk.comp_measurable hf.measurable_mk,
(ae_eq_comp hf hg.ae_eq_mk).trans (hf.ae_eq_mk.fun_comp (hg.mk g))⟩
#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
+-/
+#print MeasureTheory.AEStronglyMeasurable.comp_measurable /-
theorem comp_measurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
AEStronglyMeasurable (g ∘ f) μ :=
hg.comp_aemeasurable hf.AEMeasurable
#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurable
+-/
+#print MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving /-
theorem comp_quasiMeasurePreserving {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α}
{f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
(hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
(hg.mono' hf.AbsolutelyContinuous).comp_measurable hf.Measurable
#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
+-/
+#print MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range /-
theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
@@ -1713,6 +1910,7 @@ theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
filter_upwards [hf.ae_eq_mk] with x hx
simp [hx]
#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
+-/
#print aestronglyMeasurable_iff_aemeasurable_separable /-
/-- A function is almost everywhere strongly measurable if and only if it is almost everywhere
@@ -1734,6 +1932,7 @@ theorem aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace
#align ae_strongly_measurable_iff_ae_measurable_separable aestronglyMeasurable_iff_aemeasurable_separable
-/
+#print MeasurableEmbedding.aestronglyMeasurable_map_iff /-
theorem MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _} {mγ : MeasurableSpace γ}
{mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ} (hf : MeasurableEmbedding f) {g : α → β} :
AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ :=
@@ -1743,7 +1942,9 @@ theorem MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _} {mγ : Me
rcases hf.exists_strongly_measurable_extend hgm₁ fun x => ⟨g x⟩ with ⟨g₂, hgm₂, rfl⟩
exact ⟨g₂, hgm₂, hf.ae_map_iff.2 HEq⟩
#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aestronglyMeasurable_map_iff
+-/
+#print Embedding.aestronglyMeasurable_comp_iff /-
theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [PseudoMetrizableSpace γ]
{g : β → γ} {f : α → β} (hg : Embedding g) :
AEStronglyMeasurable (fun x => g (f x)) μ ↔ AEStronglyMeasurable f μ :=
@@ -1766,13 +1967,16 @@ theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
· rcases(aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
exact ⟨g ⁻¹' t, hg.is_separable_preimage ht, h't⟩
#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iff
+-/
+#print MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff /-
theorem MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type _} {f : α → β}
{mα : MeasurableSpace α} {μa : Measure α} {mβ : MeasurableSpace β} {μb : Measure β}
(hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ} :
AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb := by
rw [← hf.map_eq, h₂.ae_strongly_measurable_map_iff]
#align measure_theory.measure_preserving.ae_strongly_measurable_comp_iff MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff
+-/
#print aestronglyMeasurable_of_tendsto_ae /-
/-- An almost everywhere sequential limit of almost everywhere strongly measurable functions is
@@ -1817,6 +2021,7 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-/
+#print MeasureTheory.AEStronglyMeasurable.sum_measure /-
theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
(h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) :=
by
@@ -1833,12 +2038,15 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
filter_upwards [ht i] with x hx
exact ⟨i, hx⟩
#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measure
+-/
+#print aestronglyMeasurable_sum_measure_iff /-
@[simp]
theorem aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : MeasurableSpace α}
{μ : ι → Measure α} : AEStronglyMeasurable f (Sum μ) ↔ ∀ i, AEStronglyMeasurable f (μ i) :=
⟨fun h i => h.mono_measure (Measure.le_sum _ _), sum_measure⟩
#align ae_strongly_measurable_sum_measure_iff aestronglyMeasurable_sum_measure_iff
+-/
#print aestronglyMeasurable_add_measure_iff /-
@[simp]
@@ -1856,12 +2064,15 @@ theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AEStronglyMeasurable.add_measure
-/
+#print MeasureTheory.AEStronglyMeasurable.iUnion /-
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
(h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
(sum_measure h).mono_measure <| restrict_iUnion_le
#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AEStronglyMeasurable.iUnion
+-/
+#print aestronglyMeasurable_iUnion_iff /-
@[simp]
theorem aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
@@ -1869,6 +2080,7 @@ theorem aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → S
⟨fun h i => h.mono_measure <| restrict_mono (subset_iUnion _ _) le_rfl,
AEStronglyMeasurable.iUnion⟩
#align ae_strongly_measurable_Union_iff aestronglyMeasurable_iUnion_iff
+-/
#print aestronglyMeasurable_union_iff /-
@[simp]
@@ -1879,6 +2091,7 @@ theorem aestronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α}
#align ae_strongly_measurable_union_iff aestronglyMeasurable_union_iff
-/
+#print MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff /-
theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
{a b : α} :
AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
@@ -1886,11 +2099,14 @@ theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
AEStronglyMeasurable f (μ.restrict <| Ioc b a) :=
by rw [uIoc_eq_union, aestronglyMeasurable_union_iff]
#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
+-/
+#print MeasureTheory.AEStronglyMeasurable.smul_measure /-
theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AEStronglyMeasurable.smul_measure
+-/
section NormedSpace
@@ -1898,10 +2114,12 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+#print aestronglyMeasurable_smul_const_iff /-
theorem aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ :=
(closedEmbedding_smul_left hc).toEmbedding.aestronglyMeasurable_comp_iff
#align ae_strongly_measurable_smul_const_iff aestronglyMeasurable_smul_const_iff
+-/
end NormedSpace
@@ -1915,21 +2133,27 @@ variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+#print aestronglyMeasurable_const_smul_iff /-
theorem aestronglyMeasurable_const_smul_iff (c : G) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iff
+-/
+#print IsUnit.aestronglyMeasurable_const_smul_iff /-
theorem IsUnit.aestronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
let ⟨u, hu⟩ := hc
hu ▸ aestronglyMeasurable_const_smul_iff u
#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aestronglyMeasurable_const_smul_iff
+-/
+#print aestronglyMeasurable_const_smul_iff₀ /-
theorem aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
(IsUnit.mk0 _ hc).aestronglyMeasurable_const_smul_iff
#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀
+-/
end MulAction
@@ -1943,24 +2167,31 @@ variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+#print StronglyMeasurable.apply_continuousLinearMap /-
theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {φ : α → F →L[𝕜] E}
(hφ : StronglyMeasurable φ) (v : F) : StronglyMeasurable fun a => φ a v :=
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_stronglyMeasurable hφ
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
+-/
+#print MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap /-
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_aestronglyMeasurable hφ
#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
+-/
+#print ContinuousLinearMap.aestronglyMeasurable_comp₂ /-
theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => L (f x) (g x)) μ :=
L.continuous₂.comp_aestronglyMeasurable <| hf.prod_mk hg
#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂
+-/
end ContinuousLinearMapNontriviallyNormedField
+#print aestronglyMeasurable_withDensity_iff /-
theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
@@ -1987,6 +2218,7 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iff
+-/
end AeStronglyMeasurable
@@ -2009,14 +2241,18 @@ protected noncomputable def mk (f : α → β) (hf : AEFinStronglyMeasurable f
#align measure_theory.ae_fin_strongly_measurable.mk MeasureTheory.AEFinStronglyMeasurable.mk
-/
+#print MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk /-
theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
FinStronglyMeasurable (hf.mk f) μ :=
hf.choose_spec.1
#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk /-
theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
+-/
#print MeasureTheory.AEFinStronglyMeasurable.aemeasurable /-
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
@@ -2030,34 +2266,44 @@ end Mk
section Arithmetic
+#print MeasureTheory.AEFinStronglyMeasurable.mul /-
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mul
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.add /-
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ :=
⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.add hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.add
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.neg /-
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
AEFinStronglyMeasurable (-f) μ :=
⟨-hf.mk f, hf.finStronglyMeasurable_mk.neg, hf.ae_eq_mk.neg⟩
#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.neg
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.sub /-
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ :=
⟨hf.mk f - hg.mk g, hf.finStronglyMeasurable_mk.sub hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.sub
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.const_smul /-
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
AEFinStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.finStronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AEFinStronglyMeasurable.const_smul
+-/
end Arithmetic
@@ -2065,22 +2311,27 @@ section Order
variable [Zero β]
+#print MeasureTheory.AEFinStronglyMeasurable.sup /-
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.sup
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.inf /-
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.inf hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AEFinStronglyMeasurable.inf
+-/
end Order
variable [Zero β] [T2Space β]
+#print MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite /-
theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict (tᶜ)] 0 ∧ SigmaFinite (μ.restrict t) :=
by
@@ -2091,6 +2342,7 @@ theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
rw [eventually_eq, ae_restrict_iff' ht.compl]
exact eventually_of_forall hgt_zero
#align measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite
+-/
#print MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet /-
/-- A measurable set `t` such that `f =ᵐ[μ.restrict tᶜ] 0` and `sigma_finite (μ.restrict t)`. -/
@@ -2099,15 +2351,19 @@ def sigmaFiniteSet (hf : AEFinStronglyMeasurable f μ) : Set α :=
#align measure_theory.ae_fin_strongly_measurable.sigma_finite_set MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet
-/
+#print MeasureTheory.AEFinStronglyMeasurable.measurableSet /-
protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
MeasurableSet hf.sigmaFiniteSet :=
hf.exists_set_sigmaFinite.choose_spec.1
#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSet
+-/
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl /-
theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
f =ᵐ[μ.restrict (hf.sigmaFiniteSetᶜ)] 0 :=
hf.exists_set_sigmaFinite.choose_spec.2.1
#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
+-/
#print MeasureTheory.AEFinStronglyMeasurable.sigmaFinite_restrict /-
instance sigmaFinite_restrict (hf : AEFinStronglyMeasurable f μ) :
@@ -2124,22 +2380,27 @@ variable {G : Type _} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : Measure
[SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G]
{f : α → G}
+#print MeasureTheory.finStronglyMeasurable_iff_measurable /-
/-- In a space with second countable topology and a sigma-finite measure, `fin_strongly_measurable`
and `measurable` are equivalent. -/
theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Measure α)
[SigmaFinite μ] : FinStronglyMeasurable f μ ↔ Measurable f :=
⟨fun h => h.Measurable, fun h => (Measurable.stronglyMeasurable h).FinStronglyMeasurable μ⟩
#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurable
+-/
+#print MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable /-
/-- In a space with second countable topology and a sigma-finite measure,
`ae_fin_strongly_measurable` and `ae_measurable` are equivalent. -/
theorem aefinStronglyMeasurable_iff_aemeasurable {m0 : MeasurableSpace α} (μ : Measure α)
[SigmaFinite μ] : AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ := by
simp_rw [ae_fin_strongly_measurable, AEMeasurable, fin_strongly_measurable_iff_measurable]
#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable
+-/
end SecondCountableTopology
+#print MeasureTheory.measurable_uncurry_of_continuous_of_measurable /-
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [TopologicalSpace ι]
[MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
{mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
@@ -2175,7 +2436,9 @@ theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [Topo
refine' h_meas.comp (Measurable.prod_mk _ measurable_snd)
exact ((t_sf n).Measurable.comp measurable_fst).subtype_mk
#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurable
+-/
+#print MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable /-
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type _}
[TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
[OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
@@ -2219,6 +2482,7 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
refine' h_str_meas.comp_measurable (Measurable.prod_mk _ measurable_snd)
exact ((t_sf n).Measurable.comp measurable_fst).subtype_mk
#align measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable
+-/
end MeasureTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -258,7 +258,6 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
· exact min_le_left _ _
· exact le_min zero_le_one (div_nonneg ((norm_nonneg _).trans hfx) (norm_nonneg _))
_ = ‖hf.approx n x‖ := by rw [norm_one, one_mul]
-
rw [← one_smul ℝ (f x)]
refine' tendsto.smul _ h_tendsto
have : min 1 (c / ‖f x‖) = 1 :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004
@@ -319,7 +319,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » t) -/
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
(ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
@@ -952,9 +952,9 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hf_zero : ∀ (x) (_ : x ∉ s), f x = 0) :
@@ -978,7 +978,7 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
exact tendsto_const_nhds
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
on `s` is `m`-strongly-measurable, then `f` is also `m₂`-strongly-measurable. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -274,7 +274,7 @@ theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [N
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : strongly_measurable[m] f) {c : ℝ}
(hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
∀ᵐ x ∂μ, Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) := by
- filter_upwards [hf_bound]with x hfx using tendsto_approx_bounded_of_norm_le hf hfx
+ filter_upwards [hf_bound] with x hfx using tendsto_approx_bounded_of_norm_le hf hfx
#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
@@ -812,7 +812,7 @@ protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s :
strongly_measurable_const`, but replacing `strongly_measurable.ite` by
`strongly_measurable.piecewise` in that example proof does not work. -/
protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α → Prop}
- {_ : DecidablePred p} (hp : MeasurableSet { a : α | p a }) (hf : StronglyMeasurable f)
+ {_ : DecidablePred p} (hp : MeasurableSet {a : α | p a}) (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable fun x => ite (p x) (f x) (g x) :=
StronglyMeasurable.piecewise hp hf hg
#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite
@@ -822,45 +822,46 @@ theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpac
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
(hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by
classical
- let f : ℕ → α →ₛ β := fun n =>
- { toFun := fun x =>
- if hx : x ∈ s then hc.approx n ⟨x, hx⟩
- else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
- measurableSet_fiber' := by
- intro x
- convert(hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
- ((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diffₓ hs)
- ext1 y
- simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
- Subtype.coe_mk, exists_and_right, exists_eq_right, mem_diff]
- by_cases hy : y ∈ s
- · rw [dif_pos hy]
- simp only [hy, exists_true_left, not_true, and_false_iff, or_false_iff]
- · rw [dif_neg hy]
- have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- simp only [A, hy, false_or_iff, IsEmpty.exists_iff, not_false_iff, and_true_iff,
- exists_true_left]
- finite_range' :=
- by
- apply ((hc.approx n).finite_range.union (hd.approx n).finite_range).Subset
- rintro - ⟨y, rfl⟩
- dsimp
- by_cases hy : y ∈ s
- · left
- rw [dif_pos hy]
- exact mem_range_self _
- · right
- rw [dif_neg hy]
- exact mem_range_self _ }
- refine' ⟨f, fun y => _⟩
- by_cases hy : y ∈ s
- · convert hc.tendsto_approx ⟨y, hy⟩ using 1
- ext1 n
- simp only [dif_pos hy, simple_func.apply_mk]
- · have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- convert hd.tendsto_approx ⟨y, A⟩ using 1
- ext1 n
- simp only [dif_neg hy, simple_func.apply_mk]
+ let f : ℕ → α →ₛ β := fun n =>
+ { toFun := fun x =>
+ if hx : x ∈ s then hc.approx n ⟨x, hx⟩
+ else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
+ measurableSet_fiber' := by
+ intro x
+ convert
+ (hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
+ ((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diffₓ hs)
+ ext1 y
+ simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
+ Subtype.coe_mk, exists_and_right, exists_eq_right, mem_diff]
+ by_cases hy : y ∈ s
+ · rw [dif_pos hy]
+ simp only [hy, exists_true_left, not_true, and_false_iff, or_false_iff]
+ · rw [dif_neg hy]
+ have A : y ∈ t := by simpa [hy] using h (mem_univ y)
+ simp only [A, hy, false_or_iff, IsEmpty.exists_iff, not_false_iff, and_true_iff,
+ exists_true_left]
+ finite_range' :=
+ by
+ apply ((hc.approx n).finite_range.union (hd.approx n).finite_range).Subset
+ rintro - ⟨y, rfl⟩
+ dsimp
+ by_cases hy : y ∈ s
+ · left
+ rw [dif_pos hy]
+ exact mem_range_self _
+ · right
+ rw [dif_neg hy]
+ exact mem_range_self _ }
+ refine' ⟨f, fun y => _⟩
+ by_cases hy : y ∈ s
+ · convert hc.tendsto_approx ⟨y, hy⟩ using 1
+ ext1 n
+ simp only [dif_pos hy, simple_func.apply_mk]
+ · have A : y ∈ t := by simpa [hy] using h (mem_univ y)
+ convert hd.tendsto_approx ⟨y, A⟩ using 1
+ ext1 n
+ simp only [dif_neg hy, simple_func.apply_mk]
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α}
@@ -930,14 +931,14 @@ theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
{f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
- MeasurableSet { x | f x = g x } := by
+ MeasurableSet {x | f x = g x} := by
borelize (E × E)
exact (hf.prod_mk hg).Measurable is_closed_diagonal.measurable_set
#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
- (hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } :=
+ (hg : StronglyMeasurable g) : MeasurableSet {a | f a < g a} :=
by
borelize (β × β)
exact (hf.prod_mk hg).Measurable is_open_lt_prod.measurable_set
@@ -945,7 +946,7 @@ theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearO
theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
- (hg : StronglyMeasurable g) : MeasurableSet { a | f a ≤ g a } :=
+ (hg : StronglyMeasurable g) : MeasurableSet {a | f a ≤ g a} :=
by
borelize (β × β)
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
@@ -1028,7 +1029,7 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
(∀ n, measurable_set[m] (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ n) ∧ (⋃ i, s i) = Set.univ :=
by
let sigma_finite_sets := spanning_sets (μ.trim hm)
- let norm_sets := fun n : ℕ => { x | ‖f x‖ ≤ n }
+ let norm_sets := fun n : ℕ => {x | ‖f x‖ ≤ n}
have norm_sets_spanning : (⋃ n, norm_sets n) = Set.univ :=
by
ext1 x; simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
@@ -1642,12 +1643,12 @@ protected theorem indicator [Zero β] (hfm : AEStronglyMeasurable f μ) {s : Set
#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun /-
theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
- NullMeasurableSet { x | f x = g x } μ :=
+ NullMeasurableSet {x | f x = g x} μ :=
by
apply
(hf.strongly_measurable_mk.measurable_set_eq_fun
hg.strongly_measurable_mk).NullMeasurableSet.congr
- filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
+ filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx
change (hf.mk f x = hg.mk g x) = (f x = g x)
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
@@ -1656,11 +1657,11 @@ theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f
#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt /-
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
- NullMeasurableSet { a | f a < g a } μ :=
+ NullMeasurableSet {a | f a < g a} μ :=
by
apply
(hf.strongly_measurable_mk.measurable_set_lt hg.strongly_measurable_mk).NullMeasurableSet.congr
- filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
+ filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx
change (hf.mk f x < hg.mk g x) = (f x < g x)
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
@@ -1669,11 +1670,11 @@ theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMe
#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le /-
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
- NullMeasurableSet { a | f a ≤ g a } μ :=
+ NullMeasurableSet {a | f a ≤ g a} μ :=
by
apply
(hf.strongly_measurable_mk.measurable_set_le hg.strongly_measurable_mk).NullMeasurableSet.congr
- filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
+ filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx
change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le
@@ -1710,7 +1711,7 @@ theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
refine' ⟨range (hf.mk f), hf.strongly_measurable_mk.is_separable_range, _⟩
- filter_upwards [hf.ae_eq_mk]with x hx
+ filter_upwards [hf.ae_eq_mk] with x hx
simp [hx]
#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
@@ -1789,7 +1790,7 @@ theorem aestronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace
(aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
choose t t_sep ht using this
refine' ⟨closure (⋃ i, t i), (is_separable_Union fun i => t_sep i).closure, _⟩
- filter_upwards [ae_all_iff.2 ht, limUnder]with x hx h'x
+ filter_upwards [ae_all_iff.2 ht, limUnder] with x hx h'x
apply mem_closure_of_tendsto (h'x.comp hv)
apply eventually_of_forall fun n => _
apply mem_Union_of_mem n
@@ -1812,7 +1813,7 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).AEMeasurable) h_ae_tendsto
have Hg : ae_strongly_measurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.strongly_measurable_mk, _⟩
- filter_upwards [hg, Hg.ae_eq_mk]with x hx h'x
+ filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x
rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-/
@@ -1830,7 +1831,7 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
refine' ⟨⋃ i, t i, is_separable_Union t_sep, _⟩
simp only [measure.ae_sum_eq, mem_Union, eventually_supr]
intro i
- filter_upwards [ht i]with x hx
+ filter_upwards [ht i] with x hx
exact ⟨i, hx⟩
#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measure
@@ -1968,21 +1969,21 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
by
constructor
· rintro ⟨g', g'meas, hg'⟩
- have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurable_set_singleton 0)).compl
+ have A : MeasurableSet {x : α | f x ≠ 0} := (hf (measurable_set_singleton 0)).compl
refine' ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.strongly_measurable.smul g'meas, _⟩
- apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ { x | f x ≠ 0 }
+ apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ {x | f x ≠ 0}
· rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
- filter_upwards [hg']with a ha h'a
+ filter_upwards [hg'] with a ha h'a
have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, ENNReal.coe_eq_zero] using h'a
rw [ha this]
- · filter_upwards [ae_restrict_mem A.compl]with x hx
+ · filter_upwards [ae_restrict_mem A.compl] with x hx
simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
simp [hx]
· rintro ⟨g', g'meas, hg'⟩
refine' ⟨fun x => (f x : ℝ)⁻¹ • g' x, hf.coe_nnreal_real.inv.strongly_measurable.smul g'meas, _⟩
rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal]
- filter_upwards [hg']with x hx h'x
+ filter_upwards [hg'] with x hx h'x
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -75,7 +75,7 @@ open scoped ENNReal Topology MeasureTheory NNReal BigOperators
the two spaces has second countable topology. This is the right assumption to ensure that continuous
maps from `α` to `β` are strongly measurable. -/
class SecondCountableTopologyEither (α β : Type _) [TopologicalSpace α] [TopologicalSpace β] :
- Prop where
+ Prop where
out : SecondCountableTopology α ∨ SecondCountableTopology β
#align second_countable_topology_either SecondCountableTopologyEither
-/
@@ -240,8 +240,8 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
have h_tendsto := hf.tendsto_approx x
simp only [strongly_measurable.approx_bounded, simple_func.coe_map, Function.comp_apply]
by_cases hfx0 : ‖f x‖ = 0
- · rw [norm_eq_zero] at hfx0
- rw [hfx0] at h_tendsto⊢
+ · rw [norm_eq_zero] at hfx0
+ rw [hfx0] at h_tendsto ⊢
have h_tendsto_norm : tendsto (fun n => ‖hf.approx n x‖) at_top (𝓝 0) :=
by
convert h_tendsto.norm
@@ -309,7 +309,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
have : ∀ n, ∃ c, ∀ x, fs n x = c := fun n => simple_func.simple_func_bot (fs n)
let cs n := (this n).some
have h_cs_eq : ∀ n, ⇑(fs n) = fun x => cs n := fun n => funext (this n).choose_spec
- simp_rw [h_cs_eq] at h_fs_tendsto
+ simp_rw [h_cs_eq] at h_fs_tendsto
have h_tendsto : tendsto cs at_top (𝓝 (f hα.some)) := h_fs_tendsto hα.some
ext1 x
exact tendsto_nhds_unique (h_fs_tendsto x) h_tendsto
@@ -341,10 +341,10 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
refine' fun n => (measure_bUnion_finset_le _ _).trans_lt _
refine' ennreal.sum_lt_top_iff.mpr fun y hy => _
rw [simple_func.restrict_preimage_singleton _ ((hS_meas n).inter ht)]
- swap; · rw [Finset.mem_filter] at hy; exact hy.2
+ swap; · rw [Finset.mem_filter] at hy ; exact hy.2
refine' (measure_mono (Set.inter_subset_left _ _)).trans_lt _
have h_lt_top := measure_spanning_sets_lt_top (μ.restrict t) n
- rwa [measure.restrict_apply' ht] at h_lt_top
+ rwa [measure.restrict_apply' ht] at h_lt_top
· by_cases hxt : x ∈ t
swap; · rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]; exact tendsto_const_nhds
have h : tendsto (fun n => (f_approx n) x) at_top (𝓝 (f x)) := hf_meas.tendsto_approx x
@@ -361,7 +361,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
refine' ⟨n, fun m hnm => _⟩
simp_rw [fs, simple_func.restrict_apply _ ((hS_meas m).inter ht),
Set.indicator_of_mem (hn m hnm)]
- rw [tendsto_at_top'] at h⊢
+ rw [tendsto_at_top'] at h ⊢
intro s hs
obtain ⟨n₂, hn₂⟩ := h s hs
refine' ⟨max n₁ n₂, fun m hm => _⟩
@@ -597,7 +597,7 @@ include m
theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable l.Prod := by
induction' l with f l ihl; · exact strongly_measurable_one
- rw [List.forall_mem_cons] at hl
+ rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
@@ -724,7 +724,7 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
by
apply ClosedEmbedding.measurableEmbedding
exact closedEmbedding_subtype_val isClosed_closure
- have g_meas : Measurable g := by rw [fg] at H; exact T.measurable_comp_iff.1 H
+ have g_meas : Measurable g := by rw [fg] at H ; exact T.measurable_comp_iff.1 H
have : second_countable_topology (closure (range f)) :=
by
suffices separable_space (closure (range f)) by
@@ -789,7 +789,7 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
(is_separable_Union fun i => (hf (v i)).isSeparable_range).closure
apply this.mono
rintro _ ⟨x, rfl⟩
- rw [tendsto_pi_nhds] at lim
+ rw [tendsto_pi_nhds] at lim
apply mem_closure_of_tendsto ((limUnder x).comp hv)
apply eventually_of_forall fun n => _
apply mem_Union_of_mem n
@@ -1123,7 +1123,7 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
· have h_fs_zero : ∀ n, ∀ x ∈ tᶜ, fs n x = 0 :=
by
intro n x hxt
- rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
+ rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
simpa using hxt n
refine' fun x hxt => tendsto_nhds_unique (h_approx x) _
rw [funext fun n => h_fs_zero n x hxt]
@@ -1492,7 +1492,7 @@ theorem List.aestronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
by
induction' l with f l ihl; · exact ae_strongly_measurable_one
- rw [List.forall_mem_cons] at hl
+ rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
@@ -1724,7 +1724,7 @@ theorem aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace
refine' ⟨fun H => ⟨H.AEMeasurable, H.isSeparable_ae_range⟩, _⟩
rintro ⟨H, ⟨t, t_sep, ht⟩⟩
rcases eq_empty_or_nonempty t with (rfl | h₀)
- · simp only [mem_empty_iff_false, eventually_false_iff_eq_bot, ae_eq_bot] at ht
+ · simp only [mem_empty_iff_false, eventually_false_iff_eq_bot, ae_eq_bot] at ht
rw [ht]
exact ae_strongly_measurable_zero_measure f
· obtain ⟨g, g_meas, gt, fg⟩ : ∃ g : α → β, Measurable g ∧ range g ⊆ t ∧ f =ᵐ[μ] g :=
@@ -1803,17 +1803,17 @@ one can select a strongly measurable function as the almost everywhere limit. -/
theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β] {f : ℕ → α → β}
(hf : ∀ n, AEStronglyMeasurable (f n) μ)
(h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
- ∃ (f_lim : α → β)(hf_lim_meas : StronglyMeasurable f_lim),
+ ∃ (f_lim : α → β) (hf_lim_meas : StronglyMeasurable f_lim),
∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) :=
by
borelize β
obtain ⟨g, g_meas, hg⟩ :
- ∃ (g : α → β)(g_meas : Measurable g), ∀ᵐ x ∂μ, tendsto (fun n => f n x) at_top (𝓝 (g x)) :=
+ ∃ (g : α → β) (g_meas : Measurable g), ∀ᵐ x ∂μ, tendsto (fun n => f n x) at_top (𝓝 (g x)) :=
measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).AEMeasurable) h_ae_tendsto
have Hg : ae_strongly_measurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.strongly_measurable_mk, _⟩
filter_upwards [hg, Hg.ae_eq_mk]with x hx h'x
- rwa [h'x] at hx
+ rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-/
@@ -1971,20 +1971,20 @@ theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurable_set_singleton 0)).compl
refine' ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.strongly_measurable.smul g'meas, _⟩
apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ { x | f x ≠ 0 }
- · rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
+ · rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
filter_upwards [hg']with a ha h'a
have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, ENNReal.coe_eq_zero] using h'a
rw [ha this]
· filter_upwards [ae_restrict_mem A.compl]with x hx
- simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
+ simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
simp [hx]
· rintro ⟨g', g'meas, hg'⟩
refine' ⟨fun x => (f x : ℝ)⁻¹ • g' x, hf.coe_nnreal_real.inv.strongly_measurable.smul g'meas, _⟩
rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal]
filter_upwards [hg']with x hx h'x
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
- simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
+ simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iff
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -1920,11 +1920,11 @@ theorem aestronglyMeasurable_const_smul_iff (c : G) :
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iff
-theorem IsUnit.aEStronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
+theorem IsUnit.aestronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
let ⟨u, hu⟩ := hc
hu ▸ aestronglyMeasurable_const_smul_iff u
-#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aEStronglyMeasurable_const_smul_iff
+#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aestronglyMeasurable_const_smul_iff
theorem aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -68,7 +68,7 @@ measurable functions, as a basis for the Bochner integral.
open MeasureTheory Filter TopologicalSpace Function Set MeasureTheory.Measure
-open ENNReal Topology MeasureTheory NNReal BigOperators
+open scoped ENNReal Topology MeasureTheory NNReal BigOperators
#print SecondCountableTopologyEither /-
/-- The typeclass `second_countable_topology_either α β` registers the fact that at least one of
@@ -141,7 +141,7 @@ def AEFinStronglyMeasurable [Zero β] {m0 : MeasurableSpace α} (f : α → β)
end Definitions
-open MeasureTheory
+open scoped MeasureTheory
/-! ## Strongly measurable functions -/
@@ -562,7 +562,7 @@ variable [MeasurableSpace α] [TopologicalSpace β]
open Filter
-open Filter
+open scoped Filter
#print MeasureTheory.StronglyMeasurable.sup /-
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
@@ -1653,6 +1653,7 @@ theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f
#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
-/
+#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt /-
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a < g a } μ :=
@@ -1663,7 +1664,9 @@ theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMe
change (hf.mk f x < hg.mk g x) = (f x < g x)
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
+-/
+#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le /-
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ :=
@@ -1674,6 +1677,7 @@ theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetri
change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le
+-/
#print aestronglyMeasurable_of_aestronglyMeasurable_trim /-
theorem aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -146,24 +146,12 @@ open MeasureTheory
/-! ## Strongly measurable functions -/
-/- warning: measure_theory.strongly_measurable.ae_strongly_measurable -> MeasureTheory.StronglyMeasurable.aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {μ : MeasureTheory.Measure.{u1} α m0}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m0 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m0 f μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m0 : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {μ : MeasureTheory.Measure.{u2} α m0}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m0 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m0 f μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurableₓ'. -/
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {m0 : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
AEStronglyMeasurable f μ :=
⟨f, hf, EventuallyEq.refl _ _⟩
#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurable
-/- warning: measure_theory.subsingleton.strongly_measurable -> MeasureTheory.Subsingleton.stronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : Subsingleton.{succ u2} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : Subsingleton.{succ u1} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f
-Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurableₓ'. -/
@[simp]
theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton β] (f : α → β) : StronglyMeasurable f :=
@@ -175,45 +163,21 @@ theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [Topologica
exact MeasurableSet.univ
#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable
-/- warning: measure_theory.simple_func.strongly_measurable -> MeasureTheory.SimpleFunc.stronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α m β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α m β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f)
-Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurableₓ'. -/
theorem SimpleFunc.stronglyMeasurable {α β} {m : MeasurableSpace α} [TopologicalSpace β]
(f : α →ₛ β) : StronglyMeasurable f :=
⟨fun _ => f, fun x => tendsto_const_nhds⟩
#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable
-/- warning: measure_theory.strongly_measurable_of_is_empty -> MeasureTheory.stronglyMeasurable_of_isEmpty is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : IsEmpty.{succ u1} α] {m : MeasurableSpace.{u1} α} [_inst_3 : TopologicalSpace.{u2} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 m f
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : IsEmpty.{succ u2} α] {m : MeasurableSpace.{u2} α} [_inst_3 : TopologicalSpace.{u1} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 m f
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmptyₓ'. -/
theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {m : MeasurableSpace α} [TopologicalSpace β]
(f : α → β) : StronglyMeasurable f :=
⟨fun n => SimpleFunc.ofIsEmpty, isEmptyElim⟩
#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
-/- warning: measure_theory.strongly_measurable_const -> MeasureTheory.stronglyMeasurable_const is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {b : β}, MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (a : α) => b)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {b : β}, MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (fun (a : α) => b)
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_constₓ'. -/
theorem stronglyMeasurable_const {α β} {m : MeasurableSpace α} [TopologicalSpace β] {b : β} :
StronglyMeasurable fun a : α => b :=
⟨fun n => SimpleFunc.const α b, fun a => tendsto_const_nhds⟩
#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_const
-/- warning: measure_theory.strongly_measurable_one -> MeasureTheory.stronglyMeasurable_one is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : One.{u2} β], MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 1 (OfNat.mk.{max u1 u2} (α -> β) 1 (One.one.{max u1 u2} (α -> β) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : One.{u1} β], MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 1 (One.toOfNat1.{max u2 u1} (α -> β) (Pi.instOne.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.1583 : α) => β) (fun (i : α) => _inst_3))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_oneₓ'. -/
@[to_additive]
theorem stronglyMeasurable_one {α β} {m : MeasurableSpace α} [TopologicalSpace β] [One β] :
StronglyMeasurable (1 : α → β) :=
@@ -221,12 +185,6 @@ theorem stronglyMeasurable_one {α β} {m : MeasurableSpace α} [TopologicalSpac
#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_one
#align measure_theory.strongly_measurable_zero MeasureTheory.stronglyMeasurable_zero
-/- warning: measure_theory.strongly_measurable_const' -> MeasureTheory.stronglyMeasurable_const' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (forall (x : α) (y : α), Eq.{succ u2} β (f x) (f y)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (forall (x : α) (y : α), Eq.{succ u1} β (f x) (f y)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f)
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'ₓ'. -/
/-- A version of `strongly_measurable_const` that assumes `f x = f y` for all `x, y`.
This version works for functions between empty types. -/
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
@@ -237,12 +195,6 @@ theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalS
· convert strongly_measurable_const; exact funext fun x => hf x h.some
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
-/- warning: measure_theory.subsingleton.strongly_measurable' -> MeasureTheory.Subsingleton.stronglyMeasurable' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : Subsingleton.{succ u1} α] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : Subsingleton.{succ u2} α] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f
-Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.stronglyMeasurable'ₓ'. -/
@[simp]
theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton α] (f : α → β) : StronglyMeasurable f :=
@@ -266,12 +218,6 @@ protected noncomputable def approx {m : MeasurableSpace α} (hf : StronglyMeasur
#align measure_theory.strongly_measurable.approx MeasureTheory.StronglyMeasurable.approx
-/
-/- warning: measure_theory.strongly_measurable.tendsto_approx -> MeasureTheory.StronglyMeasurable.tendsto_approx is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (x : α), Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approx.{u1, u2} α β f _inst_2 m hf n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β _inst_2 (f x))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] {m : MeasurableSpace.{u2} α} (hf : MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (x : α), Filter.Tendsto.{0, u1} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β (MeasureTheory.StronglyMeasurable.approx.{u2, u1} α β f _inst_2 m hf n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} β _inst_2 (f x))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx MeasureTheory.StronglyMeasurable.tendsto_approxₓ'. -/
protected theorem tendsto_approx {m : MeasurableSpace α} (hf : StronglyMeasurable f) :
∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
hf.choose_spec
@@ -287,9 +233,6 @@ noncomputable def approxBounded {m : MeasurableSpace α} [Norm β] [SMul ℝ β]
#align measure_theory.strongly_measurable.approx_bounded MeasureTheory.StronglyMeasurable.approxBounded
-/
-/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_leₓ'. -/
theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} (hf : strongly_measurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) :=
@@ -327,9 +270,6 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
refine' tendsto.div tendsto_const_nhds h_tendsto.norm hfx0
#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
-/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_ae -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_aeₓ'. -/
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : strongly_measurable[m] f) {c : ℝ}
(hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
@@ -337,9 +277,6 @@ theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [N
filter_upwards [hf_bound]with x hfx using tendsto_approx_bounded_of_norm_le hf hfx
#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
-/- warning: measure_theory.strongly_measurable.norm_approx_bounded_le -> MeasureTheory.StronglyMeasurable.norm_approxBounded_le is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_leₓ'. -/
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : strongly_measurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
‖hf.approxBounded c n x‖ ≤ c :=
@@ -360,12 +297,6 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
· rwa [div_le_one (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]
#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_le
-/- warning: strongly_measurable_bot_iff -> stronglyMeasurable_bot_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Nonempty.{succ u2} β] [_inst_4 : T2Space.{u2} β _inst_2], Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 (Bot.bot.{u1} (MeasurableSpace.{u1} α) (CompleteLattice.toHasBot.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.completeLattice.{u1} α))) f) (Exists.{succ u2} β (fun (c : β) => Eq.{max (succ u1) (succ u2)} (α -> β) f (fun (_x : α) => c)))
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-Case conversion may be inaccurate. Consider using '#align strongly_measurable_bot_iff stronglyMeasurable_bot_iffₓ'. -/
theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
strongly_measurable[⊥] f ↔ ∃ c, f = fun _ => c :=
by
@@ -388,12 +319,6 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- warning: measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite -> MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {t : Set.{u1} α}, (MeasurableSet.{u1} α m t) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_3))))) -> (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t)) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFiniteₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
@@ -454,12 +379,6 @@ protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : Me
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
-/
-/- warning: measure_theory.strongly_measurable.measurable -> MeasureTheory.StronglyMeasurable.measurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (Measurable.{u1, u2} α β m _inst_4 f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (Measurable.{u2, u1} α β m _inst_4 f)
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurableₓ'. -/
/-- A strongly measurable function is measurable. -/
protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f :=
@@ -467,12 +386,6 @@ protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [Pse
(tendsto_pi_nhds.mpr hf.tendsto_approx)
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
-/- warning: measure_theory.strongly_measurable.ae_measurable -> MeasureTheory.StronglyMeasurable.aemeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4] {μ : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (AEMeasurable.{u1, u2} α β _inst_4 m f μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4] {μ : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (AEMeasurable.{u2, u1} α β _inst_4 m f μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurableₓ'. -/
/-- A strongly measurable function is almost everywhere measurable. -/
protected theorem aemeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
@@ -480,24 +393,12 @@ protected theorem aemeasurable {m : MeasurableSpace α} [TopologicalSpace β]
hf.Measurable.AEMeasurable
#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurable
-/- warning: continuous.comp_strongly_measurable -> Continuous.comp_stronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)))
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u1} β γ _inst_2 _inst_3 g) -> (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u3, u1} α γ _inst_3 m (fun (x : α) => g (f x)))
-Case conversion may be inaccurate. Consider using '#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurableₓ'. -/
theorem Continuous.comp_stronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => g (f x) :=
⟨fun n => SimpleFunc.map g (hf.approx n), fun x => (hg.Tendsto _).comp (hf.tendsto_approx x)⟩
#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurable
-/- warning: measure_theory.strongly_measurable.measurable_set_mul_support -> MeasureTheory.StronglyMeasurable.measurableSet_mulSupport is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : One.{u2} β] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : TopologicalSpace.MetrizableSpace.{u2} β _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 m f) -> (MeasurableSet.{u1} α m (Function.mulSupport.{u1, u2} α β _inst_2 f))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : One.{u1} β] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : TopologicalSpace.MetrizableSpace.{u1} β _inst_3], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 m f) -> (MeasurableSet.{u2} α m (Function.mulSupport.{u2, u1} α β _inst_2 f))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupportₓ'. -/
@[to_additive]
theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
[MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) := by borelize β;
@@ -505,12 +406,6 @@ theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalS
#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
#align measure_theory.strongly_measurable.measurable_set_support MeasureTheory.StronglyMeasurable.measurableSet_support
-/- warning: measure_theory.strongly_measurable.mono -> MeasureTheory.StronglyMeasurable.mono is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} {m' : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m' f) -> (LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m' m) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.monoₓ'. -/
protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
(hf : strongly_measurable[m'] f) (h_mono : m' ≤ m) : strongly_measurable[m] f :=
by
@@ -521,12 +416,6 @@ protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
exact ⟨f_approx, hf.tendsto_approx⟩
#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.mono
-/- warning: measure_theory.strongly_measurable.prod_mk -> MeasureTheory.StronglyMeasurable.prod_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_3 m g) -> (MeasureTheory.StronglyMeasurable.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u3} β γ (f x) (g x)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mkₓ'. -/
protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ]
{f : α → β} {g : α → γ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => (f x, g x) :=
@@ -536,35 +425,17 @@ protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [Topolo
exact tendsto.prod_mk (hf.tendsto_approx x) (hg.tendsto_approx x)
#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mk
-/- warning: measure_theory.strongly_measurable.comp_measurable -> MeasureTheory.StronglyMeasurable.comp_measurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {m : MeasurableSpace.{u1} α} {m' : MeasurableSpace.{u3} γ} {f : α -> β} {g : γ -> α}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (Measurable.{u3, u1} γ α m' m g) -> (MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 m' (Function.comp.{succ u3, succ u1, succ u2} γ α β f g))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurableₓ'. -/
theorem comp_measurable [TopologicalSpace β] {m : MeasurableSpace α} {m' : MeasurableSpace γ}
{f : α → β} {g : γ → α} (hf : StronglyMeasurable f) (hg : Measurable g) :
StronglyMeasurable (f ∘ g) :=
⟨fun n => SimpleFunc.comp (hf.approx n) g hg, fun x => hf.tendsto_approx (g x)⟩
#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurable
-/- warning: measure_theory.strongly_measurable.of_uncurry_left -> MeasureTheory.StronglyMeasurable.of_uncurry_left is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u3, u2} (Prod.{u1, u3} α γ) β _inst_2 (Prod.instMeasurableSpace.{u1, u3} α γ mα mγ) (Function.uncurry.{u1, u3, u2} α γ β f)) -> (forall {x : α}, MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 mγ (f x))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_leftₓ'. -/
theorem of_uncurry_left [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) :=
hf.comp_measurable measurable_prod_mk_left
#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_left
-/- warning: measure_theory.strongly_measurable.of_uncurry_right -> MeasureTheory.StronglyMeasurable.of_uncurry_right is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u3, u2} (Prod.{u1, u3} α γ) β _inst_2 (Prod.instMeasurableSpace.{u1, u3} α γ mα mγ) (Function.uncurry.{u1, u3, u2} α γ β f)) -> (forall {y : γ}, MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (fun (x : α) => f x y))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {mα : MeasurableSpace.{u2} α} {mγ : MeasurableSpace.{u1} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u2, u3} (Prod.{u2, u1} α γ) β _inst_2 (Prod.instMeasurableSpace.{u2, u1} α γ mα mγ) (Function.uncurry.{u2, u1, u3} α γ β f)) -> (forall {y : γ}, MeasureTheory.StronglyMeasurable.{u2, u3} α β _inst_2 mα (fun (x : α) => f x y))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.of_uncurry_right MeasureTheory.StronglyMeasurable.of_uncurry_rightₓ'. -/
theorem of_uncurry_right [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
StronglyMeasurable fun x => f x y :=
@@ -604,12 +475,6 @@ theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c :
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
-/
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-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {mα : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Group.{u2} β] [_inst_4 : TopologicalGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (Inv.inv.{max u1 u2} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasInv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_3))) f))
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {mα : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Group.{u2} β] [_inst_4 : TopologicalGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (Inv.inv.{max u2 u1} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => InvOneClass.toInv.{u2} β (DivInvOneMonoid.toInvOneClass.{u2} β (DivisionMonoid.toDivInvOneMonoid.{u2} β (Group.toDivisionMonoid.{u2} β _inst_3))))) f))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.invₓ'. -/
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
@@ -673,12 +538,6 @@ variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
-/- warning: strongly_measurable_const_smul_iff -> stronglyMeasurable_const_smul_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {G : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] [_inst_6 : Group.{u3} G] [_inst_7 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6))] [_inst_8 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_7)] {m : MeasurableSpace.{u1} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_7) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {f : α -> β} [G : TopologicalSpace.{u2} β] {_inst_2 : Type.{u1}} [_inst_6 : Group.{u1} _inst_2] [_inst_7 : MulAction.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6))] [_inst_8 : ContinuousConstSMul.{u1, u2} _inst_2 β G (MulAction.toSMul.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6)) _inst_7)] {m : MeasurableSpace.{u3} α} (c : _inst_2), Iff (MeasureTheory.StronglyMeasurable.{u3, u2} α β G m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} _inst_2 β β (instHSMul.{u1, u2} _inst_2 β (MulAction.toSMul.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6)) _inst_7)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β G m f)
-Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iffₓ'. -/
theorem stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
@@ -690,12 +549,6 @@ theorem IsUnit.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} {c : M
hu ▸ stronglyMeasurable_const_smul_iff u
#align is_unit.strongly_measurable_const_smul_iff IsUnit.stronglyMeasurable_const_smul_iff
-/- warning: strongly_measurable_const_smul_iff₀ -> stronglyMeasurable_const_smul_iff₀ is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {G₀ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] [_inst_9 : GroupWithZero.{u3} G₀] [_inst_10 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9))] [_inst_11 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)) _inst_10)] {m : MeasurableSpace.{u1} α} {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)))))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)) _inst_10) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u1}} {f : α -> β} [G₀ : TopologicalSpace.{u1} β] {_inst_2 : Type.{u2}} [_inst_9 : GroupWithZero.{u2} _inst_2] [_inst_10 : MulAction.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9))] [_inst_11 : ContinuousConstSMul.{u2, u1} _inst_2 β G₀ (MulAction.toSMul.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9)) _inst_10)] {m : MeasurableSpace.{u3} α} {c : _inst_2}, (Ne.{succ u2} _inst_2 c (OfNat.ofNat.{u2} _inst_2 0 (Zero.toOfNat0.{u2} _inst_2 (MonoidWithZero.toZero.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α β G₀ m (fun (x : α) => HSMul.hSMul.{u2, u1, u1} _inst_2 β β (instHSMul.{u2, u1} _inst_2 β (MulAction.toSMul.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9)) _inst_10)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u1} α β G₀ m f))
-Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀ₓ'. -/
theorem stronglyMeasurable_const_smul_iff₀ {m : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
(IsUnit.mk0 _ hc).stronglyMeasurable_const_smul_iff
@@ -740,12 +593,6 @@ variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : Mea
include m
-/- warning: list.strongly_measurable_prod' -> List.stronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : Monoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))] {m : MeasurableSpace.{u1} α} (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (List.prod.{max u1 u2} (α -> M) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))) l))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : Monoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))] {m : MeasurableSpace.{u2} α} (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (List.prod.{max u2 u1} (α -> M) (Pi.instMul.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))) (Pi.instOne.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => Monoid.toOne.{u1} M _inst_2)) l))
-Case conversion may be inaccurate. Consider using '#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable l.Prod := by
@@ -756,12 +603,6 @@ theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, S
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
#align list.strongly_measurable_sum' List.stronglyMeasurable_sum'
-/- warning: list.strongly_measurable_prod -> List.stronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : Monoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))] {m : MeasurableSpace.{u1} α} (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (x : α) => List.prod.{u2} M (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2)) (MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2)) (List.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) l)))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : Monoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))] {m : MeasurableSpace.{u2} α} (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (x : α) => List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (Monoid.toOne.{u1} M _inst_2) (List.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) l)))
-Case conversion may be inaccurate. Consider using '#align list.strongly_measurable_prod List.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem List.stronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable fun x => (l.map fun f : α → M => f x).Prod := by
@@ -777,12 +618,6 @@ variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m :
include m
-/- warning: multiset.strongly_measurable_prod' -> Multiset.stronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} (l : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (Multiset.prod.{max u1 u2} (α -> M) (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) l))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} (l : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (Multiset.prod.{max u2 u1} (α -> M) (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) l))
-Case conversion may be inaccurate. Consider using '#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.Prod := by rcases l with ⟨l⟩;
@@ -790,12 +625,6 @@ theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'
#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'
-/- warning: multiset.strongly_measurable_prod -> Multiset.stronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} (s : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (x : α) => Multiset.prod.{u2} M _inst_2 (Multiset.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) s)))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} (s : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (x : α) => Multiset.prod.{u1} M _inst_2 (Multiset.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) s)))
-Case conversion may be inaccurate. Consider using '#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, StronglyMeasurable f) :
@@ -804,12 +633,6 @@ theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prod
#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum
-/- warning: finset.strongly_measurable_prod' -> Finset.stronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) s (fun (i : ι) => f i)))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) s (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i in s, f i) :=
@@ -817,12 +640,6 @@ theorem Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s :
#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'
#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'
-/- warning: finset.strongly_measurable_prod -> Finset.stronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (a : α) => Finset.prod.{u2, u3} M ι _inst_2 s (fun (i : ι) => f i a)))
-but is expected to have type
- forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (a : α) => Finset.prod.{u1, u3} M ι _inst_2 s (fun (i : ι) => f i a)))
-Case conversion may be inaccurate. Consider using '#align finset.strongly_measurable_prod Finset.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i in s, f i a := by
@@ -832,12 +649,6 @@ theorem Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s :
end CommMonoid
-/- warning: measure_theory.strongly_measurable.is_separable_range -> MeasureTheory.StronglyMeasurable.isSeparable_range is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (TopologicalSpace.IsSeparable.{u2} β _inst_2 (Set.range.{u2, succ u1} β α f))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (TopologicalSpace.IsSeparable.{u1} β _inst_2 (Set.range.{u1, succ u2} β α f))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_rangeₓ'. -/
/-- The range of a strongly measurable function is separable. -/
theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : StronglyMeasurable f) :
TopologicalSpace.IsSeparable (range f) :=
@@ -852,12 +663,6 @@ theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : S
exact mem_range_self _
#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_range
-/- warning: measure_theory.strongly_measurable.separable_space_range_union_singleton -> MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {b : β}, TopologicalSpace.SeparableSpace.{u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (Union.union.{u2} (Set.{u2} β) (Set.hasUnion.{u2} β) (Set.range.{u2, succ u1} β α f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b))) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (Union.union.{u2} (Set.{u2} β) (Set.hasUnion.{u2} β) (Set.range.{u2, succ u1} β α f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b))) _inst_2))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (forall {b : β}, TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} β (Union.union.{u1} (Set.{u1} β) (Set.instUnionSet.{u1} β) (Set.range.{u1, succ u2} β α f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) b))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (Union.union.{u1} (Set.{u1} β) (Set.instUnionSet.{u1} β) (Set.range.{u1, succ u2} β α f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) b))) _inst_2))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.separable_space_range_union_singleton MeasureTheory.StronglyMeasurable.separableSpace_range_union_singletonₓ'. -/
theorem separableSpace_range_union_singleton {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (hf : StronglyMeasurable f) {b : β} :
SeparableSpace (range f ∪ {b} : Set β) :=
@@ -904,12 +709,6 @@ theorem stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
end SecondCountableStronglyMeasurable
-/- warning: strongly_measurable_iff_measurable_separable -> stronglyMeasurable_iff_measurable_separable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4], Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (And (Measurable.{u1, u2} α β m _inst_4 f) (TopologicalSpace.IsSeparable.{u2} β _inst_2 (Set.range.{u2, succ u1} β α f)))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4], Iff (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (And (Measurable.{u2, u1} α β m _inst_4 f) (TopologicalSpace.IsSeparable.{u1} β _inst_2 (Set.range.{u1, succ u2} β α f)))
-Case conversion may be inaccurate. Consider using '#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separableₓ'. -/
/-- A function is strongly measurable if and only if it is measurable and has separable
range. -/
theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [TopologicalSpace β]
@@ -936,12 +735,6 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
exact continuous_subtype_coe.comp_strongly_measurable g_smeas
#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separable
-/- warning: continuous.strongly_measurable -> Continuous.stronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : OpensMeasurableSpace.{u1} α _inst_3 _inst_2] {β : Type.{u2}} [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_5] [h : SecondCountableTopologyEither.{u1, u2} α β _inst_3 _inst_5] {f : α -> β}, (Continuous.{u1, u2} α β _inst_3 _inst_5 f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_5 _inst_2 f)
-but is expected to have type
- forall {α : Type.{u2}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : OpensMeasurableSpace.{u2} α _inst_3 _inst_2] {β : Type.{u1}} [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_5] [h : SecondCountableTopologyEither.{u2, u1} α β _inst_3 _inst_5] {f : α -> β}, (Continuous.{u2, u1} α β _inst_3 _inst_5 f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_5 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align continuous.strongly_measurable Continuous.stronglyMeasurableₓ'. -/
/-- A continuous function is strongly measurable when either the source space or the target space
is second-countable. -/
theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
@@ -957,12 +750,6 @@ theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
· exact hf.measurable.strongly_measurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
-/- warning: embedding.comp_strongly_measurable_iff -> Embedding.comp_stronglyMeasurable_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : TopologicalSpace.{u3} γ] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u3} γ _inst_4] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u3} β γ _inst_2 _inst_4 g) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_4 m (fun (x : α) => g (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : TopologicalSpace.{u1} γ] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} γ _inst_4] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u1} β γ _inst_2 _inst_4 g) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α γ _inst_4 m (fun (x : α) => g (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f))
-Case conversion may be inaccurate. Consider using '#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iffₓ'. -/
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
@@ -989,12 +776,6 @@ theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [Topologi
simp [hg.inj.eq_iff]
#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff
-/- warning: strongly_measurable_of_tendsto -> stronglyMeasurable_of_tendsto is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] (u : Filter.{u3} ι) [_inst_4 : Filter.NeBot.{u3} ι u] [_inst_5 : Filter.IsCountablyGenerated.{u3} ι u] {f : ι -> α -> β} {g : α -> β}, (forall (i : ι), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (f i)) -> (Filter.Tendsto.{u3, max u1 u2} ι (α -> β) f u (nhds.{max u1 u2} (α -> β) (Pi.topologicalSpace.{u1, u2} α (fun (ᾰ : α) => β) (fun (a : α) => _inst_2)) g)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendstoₓ'. -/
/-- A sequential limit of strongly measurable functions is strongly measurable. -/
theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β}
@@ -1015,12 +796,6 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
exact mem_range_self _
#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendsto
-/- warning: measure_theory.strongly_measurable.piecewise -> MeasureTheory.StronglyMeasurable.piecewise is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewiseₓ'. -/
protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
{_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) :=
@@ -1031,12 +806,6 @@ protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s :
· simpa [hx] using hg.tendsto_approx x
#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewise
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.iteₓ'. -/
/-- this is slightly different from `strongly_measurable.piecewise`. It can be used to show
`strongly_measurable (ite (x=0) 0 1)` by
`exact strongly_measurable.ite (measurable_set_singleton 0) strongly_measurable_const
@@ -1048,12 +817,6 @@ protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α →
StronglyMeasurable.piecewise hp hf hg
#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite
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theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
@@ -1100,12 +863,6 @@ theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpac
simp only [dif_neg hy, simple_func.apply_mk]
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
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theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
(h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
@@ -1114,80 +871,38 @@ theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α
h₂
#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl
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protected theorem indicator {m : MeasurableSpace α} [TopologicalSpace β] [Zero β]
(hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
StronglyMeasurable (s.indicator f) :=
hf.piecewise hs stronglyMeasurable_const
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
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protected theorem dist {m : MeasurableSpace α} {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist
-/- warning: measure_theory.strongly_measurable.norm -> MeasureTheory.StronglyMeasurable.norm is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.normₓ'. -/
protected theorem norm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ :=
continuous_norm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm
-/- warning: measure_theory.strongly_measurable.nnnorm -> MeasureTheory.StronglyMeasurable.nnnorm is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnormₓ'. -/
protected theorem nnnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ :=
continuous_nnnorm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm
-/- warning: measure_theory.strongly_measurable.ennnorm -> MeasureTheory.StronglyMeasurable.ennnorm is a dubious translation:
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- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {β : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (fun (a : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_2)) (f a))))
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnormₓ'. -/
protected theorem ennnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
(ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).Measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
-/- warning: measure_theory.strongly_measurable.real_to_nnreal -> MeasureTheory.StronglyMeasurable.real_toNNReal is a dubious translation:
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- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {f : α -> Real}, (MeasureTheory.StronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => Real.toNNReal (f x)))
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNRealₓ'. -/
protected theorem real_toNNReal {m : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => (f x).toNNReal :=
continuous_real_toNNReal.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal
-/- warning: measurable_embedding.strongly_measurable_extend -> MeasurableEmbedding.stronglyMeasurable_extend is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extendₓ'. -/
theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
@@ -1204,12 +919,6 @@ theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α
hg'.tendsto_approx x
#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
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theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
@@ -1219,12 +928,6 @@ theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g
funext fun x => hg.Injective.extend_apply _ _ _⟩
#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_funₓ'. -/
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
{f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
MeasurableSet { x | f x = g x } := by
@@ -1232,12 +935,6 @@ theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [
exact (hf.prod_mk hg).Measurable is_closed_diagonal.measurable_set
#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_ltₓ'. -/
theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } :=
@@ -1246,12 +943,6 @@ theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearO
exact (hf.prod_mk hg).Measurable is_open_lt_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_lt
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_leₓ'. -/
theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet { a | f a ≤ g a } :=
@@ -1260,12 +951,6 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_setₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
@@ -1292,12 +977,6 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
exact tendsto_const_nhds
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/- warning: measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on -> MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_onₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
@@ -1340,12 +1019,6 @@ theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableS
exact hg_seq_tendsto x
#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
-/- warning: measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le -> MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_leₓ'. -/
/-- If a function `f` is strongly measurable w.r.t. a sub-σ-algebra `m` and the measure is σ-finite
on `m`, then there exists spanning measurable sets with finite measure on which `f` has bounded
norm. In particular, `f` is integrable on each of those sets. -/
@@ -1387,12 +1060,6 @@ end StronglyMeasurable
/-! ## Finitely strongly measurable functions -/
-/- warning: measure_theory.fin_strongly_measurable_zero -> MeasureTheory.finStronglyMeasurable_zero is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zeroₓ'. -/
theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
[TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ :=
⟨0, by
@@ -1425,12 +1092,6 @@ protected noncomputable def approx : ℕ → α →ₛ β :=
#align measure_theory.fin_strongly_measurable.approx MeasureTheory.FinStronglyMeasurable.approx
-/
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-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approxₓ'. -/
protected theorem fin_support_approx : ∀ n, μ (support (hf.approx n)) < ∞ :=
hf.choose_spec.1
#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approx
@@ -1450,12 +1111,6 @@ protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
#align measure_theory.fin_strongly_measurable.strongly_measurable MeasureTheory.FinStronglyMeasurable.stronglyMeasurable
-/
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theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
(hf : FinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) :=
@@ -1493,12 +1148,6 @@ section Arithmetic
variable [TopologicalSpace β]
-/- warning: measure_theory.fin_strongly_measurable.mul -> MeasureTheory.FinStronglyMeasurable.mul is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mulₓ'. -/
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ :=
by
@@ -1509,12 +1158,6 @@ protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMe
exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul
-/- warning: measure_theory.fin_strongly_measurable.add -> MeasureTheory.FinStronglyMeasurable.add is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.addₓ'. -/
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ :=
⟨fun n => hf.approx n + hg.approx n, fun n =>
@@ -1524,12 +1167,6 @@ protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasura
fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add
-/- warning: measure_theory.fin_strongly_measurable.neg -> MeasureTheory.FinStronglyMeasurable.neg is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasNeg.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))) f) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => NegZeroClass.toNeg.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3))))) f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.negₓ'. -/
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
FinStronglyMeasurable (-f) μ :=
by
@@ -1539,12 +1176,6 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMe
exact hf.fin_support_approx n
#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg
-/- warning: measure_theory.fin_strongly_measurable.sub -> MeasureTheory.FinStronglyMeasurable.sub is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.subₓ'. -/
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ :=
⟨fun n => hf.approx n - hg.approx n, fun n =>
@@ -1554,12 +1185,6 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
-/- warning: measure_theory.fin_strongly_measurable.const_smul -> MeasureTheory.FinStronglyMeasurable.const_smul is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m0 f μ) -> (forall (c : 𝕜), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m0 (SMul.smul.{u3, max u1 u2} 𝕜 (α -> β) (Function.hasSMul.{u1, u3, u2} α 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6)))) c f) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m0 f μ) -> (forall (c : 𝕜), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m0 (HSMul.hSMul.{u3, max u1 u2, max u1 u2} 𝕜 (α -> β) (α -> β) (instHSMul.{u3, max u1 u2} 𝕜 (α -> β) (Pi.instSMul.{u1, u2, u3} α 𝕜 (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13158 : α) => β) (fun (i : α) => SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))))) c f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smulₓ'. -/
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
FinStronglyMeasurable (c • f) μ :=
@@ -1575,12 +1200,6 @@ section Order
variable [TopologicalSpace β] [Zero β]
-/- warning: measure_theory.fin_strongly_measurable.sup -> MeasureTheory.FinStronglyMeasurable.sup is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.supₓ'. -/
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ :=
by
@@ -1591,12 +1210,6 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
-/- warning: measure_theory.fin_strongly_measurable.inf -> MeasureTheory.FinStronglyMeasurable.inf is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.infₓ'. -/
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ :=
by
@@ -1611,12 +1224,6 @@ end Order
end FinStronglyMeasurable
-/- warning: measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite -> MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T2Space.{u2} β _inst_2] [_inst_4 : Zero.{u2} β] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Iff (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_4 m f μ) (And (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m t) (And (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_4))))) (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t))))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : T2Space.{u1} β _inst_2] [_inst_4 : Zero.{u1} β] {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m}, Iff (MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_4 m f μ) (And (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (Exists.{succ u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (MeasurableSet.{u2} α m t) (And (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) t)) -> (Eq.{succ u1} β (f x) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_4)))) (MeasureTheory.SigmaFinite.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ t))))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFiniteₓ'. -/
theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β}
[TopologicalSpace β] [T2Space β] [Zero β] {m : MeasurableSpace α} {μ : Measure α} :
FinStronglyMeasurable f μ ↔
@@ -1627,12 +1234,6 @@ theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
hf.2.choose_spec.2.2⟩
#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
-/- warning: measure_theory.ae_fin_strongly_measurable_zero -> MeasureTheory.aefinStronglyMeasurable_zero is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β], MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_2))))) μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m) [_inst_2 : Zero.{u1} β] [_inst_3 : TopologicalSpace.{u1} β], MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13622 : α) => β) (fun (i : α) => _inst_2)))) μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aefinStronglyMeasurable_zeroₓ'. -/
theorem aefinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
[TopologicalSpace β] : AEFinStronglyMeasurable (0 : α → β) μ :=
⟨0, finStronglyMeasurable_zero, EventuallyEq.rfl⟩
@@ -1641,23 +1242,11 @@ theorem aefinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Meas
/-! ## Almost everywhere strongly measurable functions -/
-/- warning: measure_theory.ae_strongly_measurable_const -> MeasureTheory.aestronglyMeasurable_const is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {b : β}, MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (a : α) => b) μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {b : β}, MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (fun (a : α) => b) μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_constₓ'. -/
theorem aestronglyMeasurable_const {α β} {m : MeasurableSpace α} {μ : Measure α}
[TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun a : α => b) μ :=
stronglyMeasurable_const.AEStronglyMeasurable
#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_const
-/- warning: measure_theory.ae_strongly_measurable_one -> MeasureTheory.aestronglyMeasurable_one is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : One.{u2} β], MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 1 (OfNat.mk.{max u1 u2} (α -> β) 1 (One.one.{max u1 u2} (α -> β) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3))))) μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : One.{u1} β], MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 1 (One.toOfNat1.{max u2 u1} (α -> β) (Pi.instOne.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13692 : α) => β) (fun (i : α) => _inst_3)))) μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_oneₓ'. -/
@[to_additive]
theorem aestronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
[One β] : AEStronglyMeasurable (1 : α → β) μ :=
@@ -1665,36 +1254,18 @@ theorem aestronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure
#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_one
#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aestronglyMeasurable_zero
-/- warning: measure_theory.subsingleton.ae_strongly_measurable -> MeasureTheory.Subsingleton.aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Subsingleton.{succ u2} β] {μ : MeasureTheory.Measure.{u1} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Subsingleton.{succ u1} β] {μ : MeasureTheory.Measure.{u2} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurableₓ'. -/
@[simp]
theorem Subsingleton.aestronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[Subsingleton β] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
(Subsingleton.stronglyMeasurable f).AEStronglyMeasurable
#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurable
-/- warning: measure_theory.subsingleton.ae_strongly_measurable' -> MeasureTheory.Subsingleton.aestronglyMeasurable' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Subsingleton.{succ u1} α] {μ : MeasureTheory.Measure.{u1} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Subsingleton.{succ u2} α] {μ : MeasureTheory.Measure.{u2} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'ₓ'. -/
@[simp]
theorem Subsingleton.aestronglyMeasurable' {m : MeasurableSpace α} [TopologicalSpace β]
[Subsingleton α] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
(Subsingleton.stronglyMeasurable' f).AEStronglyMeasurable
#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'
-/- warning: measure_theory.ae_strongly_measurable_zero_measure -> MeasureTheory.aestronglyMeasurable_zero_measure is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_2) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α _inst_2) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α _inst_2) (MeasureTheory.Measure.instZero.{u1} α _inst_2))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f (OfNat.ofNat.{u2} (MeasureTheory.Measure.{u2} α _inst_2) 0 (Zero.toOfNat0.{u2} (MeasureTheory.Measure.{u2} α _inst_2) (MeasureTheory.Measure.instZero.{u2} α _inst_2)))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measureₓ'. -/
@[simp]
theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
AEStronglyMeasurable f (0 : Measure α) :=
@@ -1704,12 +1275,6 @@ theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace
exact ⟨fun x => f default, strongly_measurable_const, rfl⟩
#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measure
-/- warning: measure_theory.simple_func.ae_strongly_measurable -> MeasureTheory.SimpleFunc.aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α m β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f) μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α m β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f) μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aestronglyMeasurableₓ'. -/
theorem SimpleFunc.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
(f : α →ₛ β) : AEStronglyMeasurable f μ :=
f.StronglyMeasurable.AEStronglyMeasurable
@@ -1730,12 +1295,6 @@ protected noncomputable def mk (f : α → β) (hf : AEStronglyMeasurable f μ)
#align measure_theory.ae_strongly_measurable.mk MeasureTheory.AEStronglyMeasurable.mk
-/
-/- warning: measure_theory.ae_strongly_measurable.strongly_measurable_mk -> MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 f hf)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 f hf)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mkₓ'. -/
theorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) :=
hf.choose_spec.1
#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk
@@ -1747,12 +1306,6 @@ theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpac
#align measure_theory.ae_strongly_measurable.measurable_mk MeasureTheory.AEStronglyMeasurable.measurable_mk
-/
-/- warning: measure_theory.ae_strongly_measurable.ae_eq_mk -> MeasureTheory.AEStronglyMeasurable.ae_eq_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 f hf)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 f hf)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mkₓ'. -/
theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk
@@ -1767,88 +1320,40 @@ protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
end Mk
-/- warning: measure_theory.ae_strongly_measurable.congr -> MeasureTheory.AEStronglyMeasurable.congr is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f g) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m g μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congrₓ'. -/
theorem congr (hf : AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AEStronglyMeasurable g μ :=
⟨hf.mk f, hf.stronglyMeasurable_mk, h.symm.trans hf.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congr
-/- warning: ae_strongly_measurable_congr -> aestronglyMeasurable_congr is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β}, (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {g : α -> β}, (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m g μ))
-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_congr aestronglyMeasurable_congrₓ'. -/
theorem aestronglyMeasurable_congr (h : f =ᵐ[μ] g) :
AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g μ :=
⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩
#align ae_strongly_measurable_congr aestronglyMeasurable_congr
-/- warning: measure_theory.ae_strongly_measurable.mono_measure -> MeasureTheory.AEStronglyMeasurable.mono_measure is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ν : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toHasLe.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f ν)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ν : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (LE.le.{u2} (MeasureTheory.Measure.{u2} α m) (Preorder.toLE.{u2} (MeasureTheory.Measure.{u2} α m) (PartialOrder.toPreorder.{u2} (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.instPartialOrder.{u2} α m))) ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f ν)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measureₓ'. -/
theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν ≤ μ) :
AEStronglyMeasurable f ν :=
⟨hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure
-/- warning: measure_theory.ae_strongly_measurable.mono' -> MeasureTheory.AEStronglyMeasurable.mono' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ν : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u1} α m ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f ν)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ν : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u2} α m ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f ν)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'ₓ'. -/
protected theorem mono' {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
AEStronglyMeasurable f ν :=
⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'
-/- warning: measure_theory.ae_strongly_measurable.mono_set -> MeasureTheory.AEStronglyMeasurable.mono_set is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ t)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α}, (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s t) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ t)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_setₓ'. -/
theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
AEStronglyMeasurable f (μ.restrict s) :=
ht.mono_measure (restrict_mono h le_rfl)
#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_set
-/- warning: measure_theory.ae_strongly_measurable.restrict -> MeasureTheory.AEStronglyMeasurable.restrict is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (forall {s : Set.{u1} α}, MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (forall {s : Set.{u2} α}, MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrictₓ'. -/
protected theorem restrict (hfm : AEStronglyMeasurable f μ) {s} :
AEStronglyMeasurable f (μ.restrict s) :=
hfm.mono_measure Measure.restrict_le_self
#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrict
-/- warning: measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk -> MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α} (h : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s)), Filter.Eventually.{u1} α (fun (x : α) => (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Eq.{succ u2} β (f x) (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m (MeasureTheory.Measure.restrict.{u1} α m μ s) _inst_2 f h x))) (MeasureTheory.Measure.ae.{u1} α m μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α} (h : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s)), Filter.Eventually.{u2} α (fun (x : α) => (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) -> (Eq.{succ u1} β (f x) (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m (MeasureTheory.Measure.restrict.{u2} α m μ s) _inst_2 f h x))) (MeasureTheory.Measure.ae.{u2} α m μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mkₓ'. -/
theorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (μ.restrict s)) :
∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x :=
ae_imp_of_ae_restrict h.ae_eq_mk
#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk
-/- warning: continuous.comp_ae_strongly_measurable -> Continuous.comp_aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u3, u2} β γ _inst_2 _inst_3 g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 m (fun (x : α) => g (f x)) μ)
-Case conversion may be inaccurate. Consider using '#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurableₓ'. -/
/-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable. -/
theorem Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
@@ -1856,12 +1361,6 @@ theorem Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg
⟨_, hg.comp_stronglyMeasurable hf.stronglyMeasurable_mk, EventuallyEq.fun_comp hf.ae_eq_mk g⟩
#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurable
-/- warning: continuous.ae_strongly_measurable -> Continuous.aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.{u1} α] [_inst_5 : OpensMeasurableSpace.{u1} α _inst_4 m] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_7 : SecondCountableTopologyEither.{u1, u2} α β _inst_4 _inst_2], (Continuous.{u1, u2} α β _inst_4 _inst_2 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_4 : TopologicalSpace.{u2} α] [_inst_5 : OpensMeasurableSpace.{u2} α _inst_4 m] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_7 : SecondCountableTopologyEither.{u2, u1} α β _inst_4 _inst_2], (Continuous.{u2, u1} α β _inst_4 _inst_2 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ)
-Case conversion may be inaccurate. Consider using '#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurableₓ'. -/
/-- A continuous function from `α` to `β` is ae strongly measurable when one of the two spaces is
second countable. -/
theorem Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
@@ -1870,24 +1369,12 @@ theorem Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSp
hf.StronglyMeasurable.AEStronglyMeasurable
#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurable
-/- warning: measure_theory.ae_strongly_measurable.prod_mk -> MeasureTheory.AEStronglyMeasurable.prod_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u3} β γ (f x) (g x)) μ)
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} α γ _inst_3 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, max u1 u2} α (Prod.{u2, u1} β γ) (instTopologicalSpaceProd.{u2, u1} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u1} β γ (f x) (g x)) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mkₓ'. -/
protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (fun x => (f x, g x)) μ :=
⟨fun x => (hf.mk f x, hg.mk g x), hf.stronglyMeasurable_mk.prod_mk hg.stronglyMeasurable_mk,
hf.ae_eq_mk.prod_mk hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk
-/- warning: measurable.ae_strongly_measurable -> Measurable.aestronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_6 : TopologicalSpace.SecondCountableTopology.{u2} β _inst_2] [_inst_7 : OpensMeasurableSpace.{u2} β _inst_2 _inst_4], (Measurable.{u1, u2} α β m _inst_4 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_6 : TopologicalSpace.SecondCountableTopology.{u1} β _inst_2] [_inst_7 : OpensMeasurableSpace.{u1} β _inst_2 _inst_4], (Measurable.{u2, u1} α β m _inst_4 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ)
-Case conversion may be inaccurate. Consider using '#align measurable.ae_strongly_measurable Measurable.aestronglyMeasurableₓ'. -/
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
theorem Measurable.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [MeasurableSpace β]
[PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β]
@@ -1925,12 +1412,6 @@ protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurab
#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AEStronglyMeasurable.const_add
-/
-/- warning: measure_theory.ae_strongly_measurable.inv -> MeasureTheory.AEStronglyMeasurable.inv is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inv.inv.{max u1 u2} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasInv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4))) f) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inv.inv.{max u2 u1} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => InvOneClass.toInv.{u2} β (DivInvOneMonoid.toInvOneClass.{u2} β (DivisionMonoid.toDivInvOneMonoid.{u2} β (Group.toDivisionMonoid.{u2} β _inst_4))))) f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.invₓ'. -/
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable f⁻¹ μ :=
@@ -1938,12 +1419,6 @@ protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurabl
#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AEStronglyMeasurable.neg
-/- warning: measure_theory.ae_strongly_measurable.div -> MeasureTheory.AEStronglyMeasurable.div is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (HDiv.hDiv.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHDiv.{max u1 u2} (α -> β) (Pi.instDiv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasDiv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4)))) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (HDiv.hDiv.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHDiv.{max u1 u2} (α -> β) (Pi.instDiv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toDiv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4)))) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.divₓ'. -/
@[to_additive]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ :=
@@ -1989,24 +1464,12 @@ end Arithmetic
section Order
-/- warning: measure_theory.ae_strongly_measurable.sup -> MeasureTheory.AEStronglyMeasurable.sup is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.supₓ'. -/
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.stronglyMeasurable_mk.sup hg.stronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.sup
-/- warning: measure_theory.ae_strongly_measurable.inf -> MeasureTheory.AEStronglyMeasurable.inf is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.inf MeasureTheory.AEStronglyMeasurable.infₓ'. -/
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.stronglyMeasurable_mk.inf hg.stronglyMeasurable_mk,
@@ -2024,12 +1487,6 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
-/- warning: list.ae_strongly_measurable_prod' -> List.aestronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : Monoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))] (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (List.prod.{max u1 u2} (α -> M) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))) l) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : Monoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))] (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (List.prod.{max u2 u1} (α -> M) (Pi.instMul.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))) (Pi.instOne.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => Monoid.toOne.{u1} M _inst_4)) l) μ)
-Case conversion may be inaccurate. Consider using '#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem List.aestronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
@@ -2041,12 +1498,6 @@ theorem List.aestronglyMeasurable_prod' (l : List (α → M))
#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
#align list.ae_strongly_measurable_sum' List.aestronglyMeasurable_sum'
-/- warning: list.ae_strongly_measurable_prod -> List.aestronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : Monoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))] (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (x : α) => List.prod.{u2} M (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4)) (MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4)) (List.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) l)) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : Monoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))] (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (x : α) => List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4)) (Monoid.toOne.{u1} M _inst_4) (List.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) l)) μ)
-Case conversion may be inaccurate. Consider using '#align list.ae_strongly_measurable_prod List.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem List.aestronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).Prod) μ := by
@@ -2060,12 +1511,6 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
-/- warning: multiset.ae_strongly_measurable_prod' -> Multiset.aestronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] (l : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (Multiset.prod.{max u1 u2} (α -> M) (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) l) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] (l : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (Multiset.prod.{max u2 u1} (α -> M) (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) l) μ)
-Case conversion may be inaccurate. Consider using '#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ := by
@@ -2073,12 +1518,6 @@ theorem Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'
#align multiset.ae_strongly_measurable_sum' Multiset.aestronglyMeasurable_sum'
-/- warning: multiset.ae_strongly_measurable_prod -> Multiset.aestronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] (s : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (x : α) => Multiset.prod.{u2} M _inst_4 (Multiset.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) s)) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] (s : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (x : α) => Multiset.prod.{u1} M _inst_4 (Multiset.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) s)) μ)
-Case conversion may be inaccurate. Consider using '#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
@@ -2087,12 +1526,6 @@ theorem Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prod
#align multiset.ae_strongly_measurable_sum Multiset.aestronglyMeasurable_sum
-/- warning: finset.ae_strongly_measurable_prod' -> Finset.aestronglyMeasurable_prod' is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) s (fun (i : ι) => f i)) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) s (fun (i : ι) => f i)) μ)
-Case conversion may be inaccurate. Consider using '#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i in s, f i) μ :=
@@ -2102,12 +1535,6 @@ theorem Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s
#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'
#align finset.ae_strongly_measurable_sum' Finset.aestronglyMeasurable_sum'
-/- warning: finset.ae_strongly_measurable_prod -> Finset.aestronglyMeasurable_prod is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (a : α) => Finset.prod.{u2, u3} M ι _inst_4 s (fun (i : ι) => f i a)) μ)
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (a : α) => Finset.prod.{u1, u3} M ι _inst_4 s (fun (i : ι) => f i a)) μ)
-Case conversion may be inaccurate. Consider using '#align finset.ae_strongly_measurable_prod Finset.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
@@ -2163,12 +1590,6 @@ protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AEStronglyMeasurable.norm
-/
-/- warning: measure_theory.ae_strongly_measurable.nnnorm -> MeasureTheory.AEStronglyMeasurable.nnnorm is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_4 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_4))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_4)) (f x)) μ)
-but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_4 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_4))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_4)) (f x)) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnormₓ'. -/
protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ :=
continuous_nnnorm.comp_aestronglyMeasurable hf
@@ -2189,12 +1610,6 @@ protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α →
#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AEStronglyMeasurable.edist
-/
-/- warning: measure_theory.ae_strongly_measurable.real_to_nnreal -> MeasureTheory.AEStronglyMeasurable.real_toNNReal is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (MeasureTheory.AEStronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => Real.toNNReal (f x)) μ)
-but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (MeasureTheory.AEStronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => Real.toNNReal (f x)) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AEStronglyMeasurable.real_toNNRealₓ'. -/
protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (f x).toNNReal) μ :=
continuous_real_toNNReal.comp_aestronglyMeasurable hf
@@ -2238,12 +1653,6 @@ theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f
#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
-/
-/- warning: measure_theory.ae_strongly_measurable.null_measurable_set_lt -> MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : LinearOrder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_4))))] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_4))))) (f a) (g a))) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : LinearOrder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_4)))))] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_4)))))) (f a) (g a))) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_ltₓ'. -/
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a < g a } μ :=
@@ -2255,12 +1664,6 @@ theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMe
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
-/- warning: measure_theory.ae_strongly_measurable.null_measurable_set_le -> MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : Preorder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 _inst_4] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LE.le.{u2} β (Preorder.toHasLe.{u2} β _inst_4) (f a) (g a))) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : Preorder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 _inst_4] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_4) (f a) (g a))) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_leₓ'. -/
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ :=
@@ -2280,12 +1683,6 @@ theorem aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : Measurabl
#align ae_strongly_measurable_of_ae_strongly_measurable_trim aestronglyMeasurable_of_aestronglyMeasurable_trim
-/
-/- warning: measure_theory.ae_strongly_measurable.comp_ae_measurable -> MeasureTheory.AEStronglyMeasurable.comp_aemeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) -> (AEMeasurable.{u3, u1} γ α mα mγ f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) -> (AEMeasurable.{u3, u2} γ α mα mγ f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurableₓ'. -/
theorem comp_aemeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
AEStronglyMeasurable (g ∘ f) μ :=
@@ -2293,36 +1690,18 @@ theorem comp_aemeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : Measur
(ae_eq_comp hf hg.ae_eq_mk).trans (hf.ae_eq_mk.fun_comp (hg.mk g))⟩
#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
-/- warning: measure_theory.ae_strongly_measurable.comp_measurable -> MeasureTheory.AEStronglyMeasurable.comp_measurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) -> (Measurable.{u3, u1} γ α mγ mα f) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) -> (Measurable.{u3, u2} γ α mγ mα f) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurableₓ'. -/
theorem comp_measurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
AEStronglyMeasurable (g ∘ f) μ :=
hg.comp_aemeasurable hf.AEMeasurable
#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurable
-/- warning: measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving -> MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ} {ν : MeasureTheory.Measure.{u1} α mα}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g ν) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u3, u1} γ α mα mγ f μ ν) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ} {ν : MeasureTheory.Measure.{u2} α mα}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g ν) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u3, u2} γ α mα mγ f μ ν) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreservingₓ'. -/
theorem comp_quasiMeasurePreserving {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α}
{f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
(hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
(hg.mono' hf.AbsolutelyContinuous).comp_measurable hf.Measurable
#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
-/- warning: measure_theory.ae_strongly_measurable.is_separable_ae_range -> MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (TopologicalSpace.IsSeparable.{u2} β _inst_2 t) (Filter.Eventually.{u1} α (fun (x : α) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (f x) t) (MeasureTheory.Measure.ae.{u1} α m μ))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (Exists.{succ u1} (Set.{u1} β) (fun (t : Set.{u1} β) => And (TopologicalSpace.IsSeparable.{u1} β _inst_2 t) (Filter.Eventually.{u2} α (fun (x : α) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (f x) t) (MeasureTheory.Measure.ae.{u2} α m μ))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AEStronglyMeasurable.isSeparable_ae_rangeₓ'. -/
theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
@@ -2351,12 +1730,6 @@ theorem aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace
#align ae_strongly_measurable_iff_ae_measurable_separable aestronglyMeasurable_iff_aemeasurable_separable
-/
-/- warning: measurable_embedding.ae_strongly_measurable_map_iff -> MeasurableEmbedding.aestronglyMeasurable_map_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasurableEmbedding.{u3, u1} γ α mγ mα f) -> (forall {g : α -> β}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasurableEmbedding.{u3, u2} γ α mγ mα f) -> (forall {g : α -> β}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ))
-Case conversion may be inaccurate. Consider using '#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aestronglyMeasurable_map_iffₓ'. -/
theorem MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _} {mγ : MeasurableSpace γ}
{mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ} (hf : MeasurableEmbedding f) {g : α → β} :
AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ :=
@@ -2367,12 +1740,6 @@ theorem MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _} {mγ : Me
exact ⟨g₂, hgm₂, hf.ae_map_iff.2 HEq⟩
#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aestronglyMeasurable_map_iff
-/- warning: embedding.ae_strongly_measurable_comp_iff -> Embedding.aestronglyMeasurable_comp_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u3} γ _inst_3] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ))
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} γ _inst_3] {g : β -> γ} {f : α -> β}, (Embedding.{u3, u2} β γ _inst_2 _inst_3 g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 m (fun (x : α) => g (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u3} α β _inst_2 m f μ))
-Case conversion may be inaccurate. Consider using '#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iffₓ'. -/
theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [PseudoMetrizableSpace γ]
{g : β → γ} {f : α → β} (hg : Embedding g) :
AEStronglyMeasurable (fun x => g (f x)) μ ↔ AEStronglyMeasurable f μ :=
@@ -2396,12 +1763,6 @@ theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
exact ⟨g ⁻¹' t, hg.is_separable_preimage ht, h't⟩
#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iff
-/- warning: measure_theory.measure_preserving.ae_strongly_measurable_comp_iff -> MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {γ : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} γ] {β : Type.{u3}} {f : α -> β} {mα : MeasurableSpace.{u1} α} {μa : MeasureTheory.Measure.{u1} α mα} {mβ : MeasurableSpace.{u3} β} {μb : MeasureTheory.Measure.{u3} β mβ}, (MeasureTheory.MeasurePreserving.{u1, u3} α β mα mβ f μa μb) -> (MeasurableEmbedding.{u1, u3} α β mα mβ f) -> (forall {g : β -> γ}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 mα (Function.comp.{succ u1, succ u3, succ u2} α β γ g f) μa) (MeasureTheory.AEStronglyMeasurable.{u3, u2} β γ _inst_3 mβ g μb))
-but is expected to have type
- forall {α : Type.{u2}} {γ : Type.{u1}} [_inst_3 : TopologicalSpace.{u1} γ] {β : Type.{u3}} {f : α -> β} {mα : MeasurableSpace.{u2} α} {μa : MeasureTheory.Measure.{u2} α mα} {mβ : MeasurableSpace.{u3} β} {μb : MeasureTheory.Measure.{u3} β mβ}, (MeasureTheory.MeasurePreserving.{u2, u3} α β mα mβ f μa μb) -> (MeasurableEmbedding.{u2, u3} α β mα mβ f) -> (forall {g : β -> γ}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α γ _inst_3 mα (Function.comp.{succ u2, succ u3, succ u1} α β γ g f) μa) (MeasureTheory.AEStronglyMeasurable.{u3, u1} β γ _inst_3 mβ g μb))
-Case conversion may be inaccurate. Consider using '#align measure_theory.measure_preserving.ae_strongly_measurable_comp_iff MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iffₓ'. -/
theorem MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type _} {f : α → β}
{mα : MeasurableSpace α} {μa : Measure α} {mβ : MeasurableSpace β} {μb : Measure β}
(hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ} :
@@ -2452,12 +1813,6 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-/
-/- warning: measure_theory.ae_strongly_measurable.sum_measure -> MeasureTheory.AEStronglyMeasurable.sum_measure is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {m : MeasurableSpace.{u1} α} {μ : ι -> (MeasureTheory.Measure.{u1} α m)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (μ i)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.sum.{u1, u3} α ι m μ))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {m : MeasurableSpace.{u2} α} {μ : ι -> (MeasureTheory.Measure.{u2} α m)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (μ i)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.sum.{u2, u1} α ι m μ))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measureₓ'. -/
theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
(h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) :=
by
@@ -2475,12 +1830,6 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
exact ⟨i, hx⟩
#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measure
-/- warning: ae_strongly_measurable_sum_measure_iff -> aestronglyMeasurable_sum_measure_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {m : MeasurableSpace.{u1} α} {μ : ι -> (MeasureTheory.Measure.{u1} α m)}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.sum.{u1, u3} α ι m μ)) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (μ i))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {m : MeasurableSpace.{u2} α} {μ : ι -> (MeasureTheory.Measure.{u2} α m)}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.sum.{u2, u1} α ι m μ)) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (μ i))
-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_sum_measure_iff aestronglyMeasurable_sum_measure_iffₓ'. -/
@[simp]
theorem aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : MeasurableSpace α}
{μ : ι → Measure α} : AEStronglyMeasurable f (Sum μ) ↔ ∀ i, AEStronglyMeasurable f (μ i) :=
@@ -2503,24 +1852,12 @@ theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AEStronglyMeasurable.add_measure
-/
-/- warning: measure_theory.ae_strongly_measurable.Union -> MeasureTheory.AEStronglyMeasurable.iUnion is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {s : ι -> (Set.{u1} α)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (s i))) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u3} α ι (fun (i : ι) => s i))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {s : ι -> (Set.{u2} α)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (s i))) -> (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AEStronglyMeasurable.iUnionₓ'. -/
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
(h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
(sum_measure h).mono_measure <| restrict_iUnion_le
#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AEStronglyMeasurable.iUnion
-/- warning: ae_strongly_measurable_Union_iff -> aestronglyMeasurable_iUnion_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {s : ι -> (Set.{u1} α)}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u3} α ι (fun (i : ι) => s i)))) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (s i)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_Union_iff aestronglyMeasurable_iUnion_iffₓ'. -/
@[simp]
theorem aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
@@ -2538,12 +1875,6 @@ theorem aestronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α}
#align ae_strongly_measurable_union_iff aestronglyMeasurable_union_iff
-/
-/- warning: measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff -> MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iffₓ'. -/
theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
{a b : α} :
AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
@@ -2552,12 +1883,6 @@ theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
by rw [uIoc_eq_union, aestronglyMeasurable_union_iff]
#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
-/- warning: measure_theory.ae_strongly_measurable.smul_measure -> MeasureTheory.AEStronglyMeasurable.smul_measure is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {R : Type.{u3}} [_inst_4 : Monoid.{u3} R] [_inst_5 : DistribMulAction.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))] [_inst_6 : IsScalarTower.{u3, 0, 0} R ENNReal ENNReal (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5))) (Mul.toSMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5)))], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (forall (c : R), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (SMul.smul.{u3, u1} R (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, u3} α R (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5))) _inst_6 m) c μ))
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- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {R : Type.{u3}} [_inst_4 : Monoid.{u3} R] [_inst_5 : DistribMulAction.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))] [_inst_6 : IsScalarTower.{u3, 0, 0} R ENNReal ENNReal (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5))) (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5)))], (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (forall (c : R), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (HSMul.hSMul.{u3, u2, u2} R (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.{u2} α m) (instHSMul.{u3, u2} R (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.instSMul.{u2, u3} α R (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5))) _inst_6 m)) c μ))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AEStronglyMeasurable.smul_measureₓ'. -/
theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
@@ -2569,9 +1894,6 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-/- warning: ae_strongly_measurable_smul_const_iff -> aestronglyMeasurable_smul_const_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_smul_const_iff aestronglyMeasurable_smul_const_iffₓ'. -/
theorem aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ :=
(closedEmbedding_smul_left hc).toEmbedding.aestronglyMeasurable_comp_iff
@@ -2589,12 +1911,6 @@ variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
-/- warning: ae_strongly_measurable_const_smul_iff -> aestronglyMeasurable_const_smul_iff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iffₓ'. -/
theorem aestronglyMeasurable_const_smul_iff (c : G) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
@@ -2606,12 +1922,6 @@ theorem IsUnit.aEStronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
hu ▸ aestronglyMeasurable_const_smul_iff u
#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aEStronglyMeasurable_const_smul_iff
-/- warning: ae_strongly_measurable_const_smul_iff₀ -> aestronglyMeasurable_const_smul_iff₀ is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀ₓ'. -/
theorem aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
(IsUnit.mk0 _ hc).aestronglyMeasurable_const_smul_iff
@@ -2629,25 +1939,16 @@ variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
-/- warning: strongly_measurable.apply_continuous_linear_map -> StronglyMeasurable.apply_continuousLinearMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {φ : α → F →L[𝕜] E}
(hφ : StronglyMeasurable φ) (v : F) : StronglyMeasurable fun a => φ a v :=
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_stronglyMeasurable hφ
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
-/- warning: measure_theory.ae_strongly_measurable.apply_continuous_linear_map -> MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_aestronglyMeasurable hφ
#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
-/- warning: continuous_linear_map.ae_strongly_measurable_comp₂ -> ContinuousLinearMap.aestronglyMeasurable_comp₂ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂ₓ'. -/
theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => L (f x) (g x)) μ :=
@@ -2656,12 +1957,6 @@ theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →
end ContinuousLinearMapNontriviallyNormedField
-/- warning: ae_strongly_measurable_with_density_iff -> aestronglyMeasurable_withDensity_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {E : Type.{u2}} [_inst_4 : NormedAddCommGroup.{u2} E] [_inst_5 : NormedSpace.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)] {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> E}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x)))) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m (fun (x : α) => SMul.smul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (Module.toMulActionWithZero.{0, u2} Real E (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4))) (NormedSpace.toModule.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4) _inst_5))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (f x)) (g x)) μ))
-but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {E : Type.{u2}} [_inst_4 : NormedAddCommGroup.{u2} E] [_inst_5 : NormedSpace.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)] {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> E}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => ENNReal.some (f x)))) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m (fun (x : α) => HSMul.hSMul.{0, u2, u2} Real E E (instHSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)) (NormedSpace.toModule.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4) _inst_5)))))) (NNReal.toReal (f x)) (g x)) μ))
-Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iffₓ'. -/
theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
@@ -2710,23 +2005,11 @@ protected noncomputable def mk (f : α → β) (hf : AEFinStronglyMeasurable f
#align measure_theory.ae_fin_strongly_measurable.mk MeasureTheory.AEFinStronglyMeasurable.mk
-/
-/- warning: measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk -> MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (MeasureTheory.AEFinStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 _inst_3 f hf) μ
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m (MeasureTheory.AEFinStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 _inst_3 f hf) μ
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mkₓ'. -/
theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
FinStronglyMeasurable (hf.mk f) μ :=
hf.choose_spec.1
#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
-/- warning: measure_theory.ae_fin_strongly_measurable.ae_eq_mk -> MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f (MeasureTheory.AEFinStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 _inst_3 f hf)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f (MeasureTheory.AEFinStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 _inst_3 f hf)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mkₓ'. -/
theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
@@ -2743,59 +2026,29 @@ end Mk
section Arithmetic
-/- warning: measure_theory.ae_fin_strongly_measurable.mul -> MeasureTheory.AEFinStronglyMeasurable.mul is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mulₓ'. -/
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mul
-/- warning: measure_theory.ae_fin_strongly_measurable.add -> MeasureTheory.AEFinStronglyMeasurable.add is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.addₓ'. -/
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ :=
⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.add hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.add
-/- warning: measure_theory.ae_fin_strongly_measurable.neg -> MeasureTheory.AEFinStronglyMeasurable.neg is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasNeg.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))) f) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => NegZeroClass.toNeg.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3))))) f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.negₓ'. -/
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
AEFinStronglyMeasurable (-f) μ :=
⟨-hf.mk f, hf.finStronglyMeasurable_mk.neg, hf.ae_eq_mk.neg⟩
#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.neg
-/- warning: measure_theory.ae_fin_strongly_measurable.sub -> MeasureTheory.AEFinStronglyMeasurable.sub is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.subₓ'. -/
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ :=
⟨hf.mk f - hg.mk g, hf.finStronglyMeasurable_mk.sub hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.sub
-/- warning: measure_theory.ae_fin_strongly_measurable.const_smul -> MeasureTheory.AEFinStronglyMeasurable.const_smul is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m f μ) -> (forall (c : 𝕜), MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m (SMul.smul.{u3, max u1 u2} 𝕜 (α -> β) (Function.hasSMul.{u1, u3, u2} α 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6)))) c f) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m f μ) -> (forall (c : 𝕜), MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m (HSMul.hSMul.{u3, max u1 u2, max u1 u2} 𝕜 (α -> β) (α -> β) (instHSMul.{u3, max u1 u2} 𝕜 (α -> β) (Pi.instSMul.{u1, u2, u3} α 𝕜 (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.21457 : α) => β) (fun (i : α) => SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))))) c f) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AEFinStronglyMeasurable.const_smulₓ'. -/
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
AEFinStronglyMeasurable (c • f) μ :=
@@ -2808,24 +2061,12 @@ section Order
variable [Zero β]
-/- warning: measure_theory.ae_fin_strongly_measurable.sup -> MeasureTheory.AEFinStronglyMeasurable.sup is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.supₓ'. -/
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.sup
-/- warning: measure_theory.ae_fin_strongly_measurable.inf -> MeasureTheory.AEFinStronglyMeasurable.inf is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
-but is expected to have type
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AEFinStronglyMeasurable.infₓ'. -/
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
@@ -2836,12 +2077,6 @@ end Order
variable [Zero β] [T2Space β]
-/- warning: measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite -> MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m t) (And (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t))) f (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))) (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t)))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ) -> (Exists.{succ u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (MeasurableSet.{u2} α m t) (And (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) t))) f (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => β) (fun (i : α) => _inst_3))))) (MeasureTheory.SigmaFinite.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ t)))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFiniteₓ'. -/
theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict (tᶜ)] 0 ∧ SigmaFinite (μ.restrict t) :=
by
@@ -2860,23 +2095,11 @@ def sigmaFiniteSet (hf : AEFinStronglyMeasurable f μ) : Set α :=
#align measure_theory.ae_fin_strongly_measurable.sigma_finite_set MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet
-/
-/- warning: measure_theory.ae_fin_strongly_measurable.measurable_set -> MeasureTheory.AEFinStronglyMeasurable.measurableSet is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), MeasurableSet.{u1} α m (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u1, u2} α β m μ _inst_2 f _inst_3 _inst_4 hf)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), MeasurableSet.{u2} α m (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u2, u1} α β m μ _inst_2 f _inst_3 _inst_4 hf)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSetₓ'. -/
protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
MeasurableSet hf.sigmaFiniteSet :=
hf.exists_set_sigmaFinite.choose_spec.1
#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSet
-/- warning: measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl -> MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u1, u2} α β m μ _inst_2 f _inst_3 _inst_4 hf)))) f (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u2, u1} α β m μ _inst_2 f _inst_3 _inst_4 hf)))) f (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => β) (fun (i : α) => _inst_3))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_complₓ'. -/
theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
f =ᵐ[μ.restrict (hf.sigmaFiniteSetᶜ)] 0 :=
hf.exists_set_sigmaFinite.choose_spec.2.1
@@ -2897,12 +2120,6 @@ variable {G : Type _} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : Measure
[SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G]
{f : α → G}
-/- warning: measure_theory.fin_strongly_measurable_iff_measurable -> MeasureTheory.finStronglyMeasurable_iff_measurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {G : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} G] [_inst_3 : MeasurableSpace.{u2} G] [_inst_4 : BorelSpace.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u1} α m0 μ], Iff (MeasureTheory.FinStronglyMeasurable.{u1, u2} α G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G (SeminormedAddGroup.toAddGroup.{u2} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} G _inst_2)))))) m0 f μ) (Measurable.{u1, u2} α G m0 _inst_3 f)
-but is expected to have type
- forall {α : Type.{u2}} {G : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} G] [_inst_3 : MeasurableSpace.{u1} G] [_inst_4 : BorelSpace.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u2} α m0 μ], Iff (MeasureTheory.FinStronglyMeasurable.{u2, u1} α G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (SubtractionCommMonoid.toSubtractionMonoid.{u1} G (AddCommGroup.toDivisionAddCommMonoid.{u1} G (SeminormedAddCommGroup.toAddCommGroup.{u1} G _inst_2)))))) m0 f μ) (Measurable.{u2, u1} α G m0 _inst_3 f)
-Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurableₓ'. -/
/-- In a space with second countable topology and a sigma-finite measure, `fin_strongly_measurable`
and `measurable` are equivalent. -/
theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Measure α)
@@ -2910,12 +2127,6 @@ theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Mea
⟨fun h => h.Measurable, fun h => (Measurable.stronglyMeasurable h).FinStronglyMeasurable μ⟩
#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurable
-/- warning: measure_theory.ae_fin_strongly_measurable_iff_ae_measurable -> MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {G : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} G] [_inst_3 : MeasurableSpace.{u2} G] [_inst_4 : BorelSpace.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u1} α m0 μ], Iff (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G (SeminormedAddGroup.toAddGroup.{u2} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} G _inst_2)))))) m0 f μ) (AEMeasurable.{u1, u2} α G _inst_3 m0 f μ)
-but is expected to have type
- forall {α : Type.{u2}} {G : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} G] [_inst_3 : MeasurableSpace.{u1} G] [_inst_4 : BorelSpace.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u2} α m0 μ], Iff (MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (SubtractionCommMonoid.toSubtractionMonoid.{u1} G (AddCommGroup.toDivisionAddCommMonoid.{u1} G (SeminormedAddCommGroup.toAddCommGroup.{u1} G _inst_2)))))) m0 f μ) (AEMeasurable.{u2, u1} α G _inst_3 m0 f μ)
-Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aefinStronglyMeasurable_iff_aemeasurableₓ'. -/
/-- In a space with second countable topology and a sigma-finite measure,
`ae_fin_strongly_measurable` and `ae_measurable` are equivalent. -/
theorem aefinStronglyMeasurable_iff_aemeasurable {m0 : MeasurableSpace α} (μ : Measure α)
@@ -2925,12 +2136,6 @@ theorem aefinStronglyMeasurable_iff_aemeasurable {m0 : MeasurableSpace α} (μ :
end SecondCountableTopology
-/- warning: measure_theory.measurable_uncurry_of_continuous_of_measurable -> MeasureTheory.measurable_uncurry_of_continuous_of_measurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u3} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u3} ι _inst_2] [_inst_4 : MeasurableSpace.{u3} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u3} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u3} ι _inst_2 _inst_4] {mβ : MeasurableSpace.{u2} β} [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : BorelSpace.{u2} β _inst_7 mβ] {m : MeasurableSpace.{u1} α} {u : ι -> α -> β}, (forall (x : α), Continuous.{u3, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), Measurable.{u1, u2} α β m mβ (u i)) -> (Measurable.{max u3 u1, u2} (Prod.{u3, u1} ι α) β (Prod.instMeasurableSpace.{u3, u1} ι α _inst_4 m) mβ (Function.uncurry.{u3, u1, u2} ι α β u))
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u1} ι _inst_2] [_inst_4 : MeasurableSpace.{u1} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u1} ι _inst_2 _inst_4] {mβ : MeasurableSpace.{u2} β} [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : BorelSpace.{u2} β _inst_7 mβ] {m : MeasurableSpace.{u3} α} {u : ι -> α -> β}, (forall (x : α), Continuous.{u1, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), Measurable.{u3, u2} α β m mβ (u i)) -> (Measurable.{max u3 u1, u2} (Prod.{u1, u3} ι α) β (Prod.instMeasurableSpace.{u1, u3} ι α _inst_4 m) mβ (Function.uncurry.{u1, u3, u2} ι α β u))
-Case conversion may be inaccurate. Consider using '#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurableₓ'. -/
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [TopologicalSpace ι]
[MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
{mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
@@ -2967,12 +2172,6 @@ theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [Topo
exact ((t_sf n).Measurable.comp measurable_fst).subtype_mk
#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurable
-/- warning: measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable -> MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u3} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u3} ι _inst_2] [_inst_4 : MeasurableSpace.{u3} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u3} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u3} ι _inst_2 _inst_4] [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : MeasurableSpace.{u1} α] {u : ι -> α -> β}, (forall (x : α), Continuous.{u3, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_7 _inst_9 (u i)) -> (MeasureTheory.StronglyMeasurable.{max u3 u1, u2} (Prod.{u3, u1} ι α) β _inst_7 (Prod.instMeasurableSpace.{u3, u1} ι α _inst_4 _inst_9) (Function.uncurry.{u3, u1, u2} ι α β u))
-but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u1} ι _inst_2] [_inst_4 : MeasurableSpace.{u1} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u1} ι _inst_2 _inst_4] [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : MeasurableSpace.{u3} α] {u : ι -> α -> β}, (forall (x : α), Continuous.{u1, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_7 _inst_9 (u i)) -> (MeasureTheory.StronglyMeasurable.{max u3 u1, u2} (Prod.{u1, u3} ι α) β _inst_7 (Prod.instMeasurableSpace.{u1, u3} ι α _inst_4 _inst_9) (Function.uncurry.{u1, u3, u2} ι α β u))
-Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurableₓ'. -/
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type _}
[TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
[OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -170,9 +170,7 @@ theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [Topologica
by
let f_sf : α →ₛ β := ⟨f, fun x => _, Set.Subsingleton.finite Set.subsingleton_of_subsingleton⟩
· exact ⟨fun n => f_sf, fun x => tendsto_const_nhds⟩
- · have h_univ : f ⁻¹' {x} = Set.univ := by
- ext1 y
- simp
+ · have h_univ : f ⁻¹' {x} = Set.univ := by ext1 y; simp
rw [h_univ]
exact MeasurableSet.univ
#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable
@@ -236,8 +234,7 @@ theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalS
by
cases isEmpty_or_nonempty α
· exact strongly_measurable_of_is_empty f
- · convert strongly_measurable_const
- exact funext fun x => hf x h.some
+ · convert strongly_measurable_const; exact funext fun x => hf x h.some
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
/- warning: measure_theory.subsingleton.strongly_measurable' -> MeasureTheory.Subsingleton.stronglyMeasurable' is a dubious translation:
@@ -419,16 +416,12 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
refine' fun n => (measure_bUnion_finset_le _ _).trans_lt _
refine' ennreal.sum_lt_top_iff.mpr fun y hy => _
rw [simple_func.restrict_preimage_singleton _ ((hS_meas n).inter ht)]
- swap
- · rw [Finset.mem_filter] at hy
- exact hy.2
+ swap; · rw [Finset.mem_filter] at hy; exact hy.2
refine' (measure_mono (Set.inter_subset_left _ _)).trans_lt _
have h_lt_top := measure_spanning_sets_lt_top (μ.restrict t) n
rwa [measure.restrict_apply' ht] at h_lt_top
· by_cases hxt : x ∈ t
- swap
- · rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]
- exact tendsto_const_nhds
+ swap; · rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]; exact tendsto_const_nhds
have h : tendsto (fun n => (f_approx n) x) at_top (𝓝 (f x)) := hf_meas.tendsto_approx x
obtain ⟨n₁, hn₁⟩ : ∃ n, ∀ m, n ≤ m → fs m x = f_approx m x :=
by
@@ -507,9 +500,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupportₓ'. -/
@[to_additive]
theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
- [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) :=
- by
- borelize β
+ [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) := by borelize β;
exact measurableSet_mulSupport hf.measurable
#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
#align measure_theory.strongly_measurable.measurable_set_support MeasureTheory.StronglyMeasurable.measurableSet_support
@@ -794,9 +785,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
- (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.Prod :=
- by
- rcases l with ⟨l⟩
+ (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.Prod := by rcases l with ⟨l⟩;
simpa using l.strongly_measurable_prod' (by simpa using hl)
#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'
#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'
@@ -931,17 +920,12 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
rintro ⟨H, H'⟩
letI := pseudo_metrizable_space_pseudo_metric β
let g := cod_restrict f (closure (range f)) fun x => subset_closure (mem_range_self x)
- have fg : f = (coe : closure (range f) → β) ∘ g :=
- by
- ext x
- rfl
+ have fg : f = (coe : closure (range f) → β) ∘ g := by ext x; rfl
have T : MeasurableEmbedding (coe : closure (range f) → β) :=
by
apply ClosedEmbedding.measurableEmbedding
exact closedEmbedding_subtype_val isClosed_closure
- have g_meas : Measurable g := by
- rw [fg] at H
- exact T.measurable_comp_iff.1 H
+ have g_meas : Measurable g := by rw [fg] at H; exact T.measurable_comp_iff.1 H
have : second_countable_topology (closure (range f)) :=
by
suffices separable_space (closure (range f)) by
@@ -1338,15 +1322,11 @@ theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableS
refine' MeasurableSet.union (hs _ (hs_m.inter _)) _
· exact @simple_func.measurable_set_fiber _ _ m _ _
by_cases hx : x = 0
- · suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = sᶜ by
- rw [this]
- exact hs_m₂.compl
+ · suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = sᶜ by rw [this]; exact hs_m₂.compl
ext1 y
rw [hx, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]
exact ⟨fun h => h.2, fun h => ⟨hg_seq_zero y h n, h⟩⟩
- · suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = ∅ by
- rw [this]
- exact MeasurableSet.empty
+ · suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = ∅ by rw [this]; exact MeasurableSet.empty
ext1 y
simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,
mem_empty_iff_false, iff_false_iff, not_and, not_not_mem]
@@ -1378,8 +1358,7 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
let norm_sets := fun n : ℕ => { x | ‖f x‖ ≤ n }
have norm_sets_spanning : (⋃ n, norm_sets n) = Set.univ :=
by
- ext1 x
- simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
+ ext1 x; simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
exact ⟨⌈‖f x‖⌉₊, Nat.le_ceil ‖f x‖⟩
let sets n := sigma_finite_sets n ∩ norm_sets n
have h_meas : ∀ n, measurable_set[m] (sets n) :=
@@ -2089,10 +2068,8 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
- (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
- by
- rcases l with ⟨l⟩
- simpa using l.ae_strongly_measurable_prod' (by simpa using hl)
+ (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ := by
+ rcases l with ⟨l⟩; simpa using l.ae_strongly_measurable_prod' (by simpa using hl)
#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'
#align multiset.ae_strongly_measurable_sum' Multiset.aestronglyMeasurable_sum'
@@ -2513,10 +2490,8 @@ theorem aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : Mea
#print aestronglyMeasurable_add_measure_iff /-
@[simp]
theorem aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
- AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν :=
- by
- rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]
- rfl
+ AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν := by
+ rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]; rfl
#align ae_strongly_measurable_add_measure_iff aestronglyMeasurable_add_measure_iff
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -291,10 +291,7 @@ noncomputable def approxBounded {m : MeasurableSpace α} [Norm β] [SMul ℝ β]
-/
/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real} {x : α}, (LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (f x)) c) -> (Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3))) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x)))
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- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real} {x : α}, (LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (NormedAddCommGroup.toNorm.{u2} β _inst_3) (f x)) c) -> (Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x)))
+<too large>
Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_leₓ'. -/
theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} (hf : strongly_measurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
@@ -334,10 +331,7 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_ae -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real}, (Filter.Eventually.{u1} α (fun (x : α) => LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (f x)) c) (MeasureTheory.Measure.ae.{u1} α m0 μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3))) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x))) (MeasureTheory.Measure.ae.{u1} α m0 μ))
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- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real}, (Filter.Eventually.{u1} α (fun (x : α) => LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (NormedAddCommGroup.toNorm.{u2} β _inst_3) (f x)) c) (MeasureTheory.Measure.ae.{u1} α m0 μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x))) (MeasureTheory.Measure.ae.{u1} α m0 μ))
+<too large>
Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_aeₓ'. -/
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : strongly_measurable[m] f) {c : ℝ}
@@ -347,10 +341,7 @@ theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [N
#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
/- warning: measure_theory.strongly_measurable.norm_approx_bounded_le -> MeasureTheory.StronglyMeasurable.norm_approxBounded_le is a dubious translation:
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- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : SeminormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField _inst_3] {m : MeasurableSpace.{u1} α} {c : Real} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m f), (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) c) -> (forall (n : Nat) (x : α), LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_3) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField _inst_3 _inst_4))))) hf c n) x)) c)
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- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : SeminormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField _inst_3] {m : MeasurableSpace.{u1} α} {c : Real} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m f), (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) c) -> (forall (n : Nat) (x : α), LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (SeminormedAddCommGroup.toNorm.{u2} β _inst_3) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m (SeminormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField _inst_3 _inst_4))))) hf c n) x)) c)
+<too large>
Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_leₓ'. -/
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : strongly_measurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
@@ -2604,10 +2595,7 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
/- warning: ae_strongly_measurable_smul_const_iff -> aestronglyMeasurable_smul_const_iff is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] [_inst_5 : CompleteSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (OfNat.mk.{u3} E 0 (Zero.zero.{u3} E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (SubNegMonoid.toAddMonoid.{u3} E (AddGroup.toSubNegMonoid.{u3} E (NormedAddGroup.toAddGroup.{u3} E (NormedAddCommGroup.toNormedAddGroup.{u3} E _inst_6)))))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) m (fun (x : α) => SMul.smul.{u2, u3} 𝕜 E (SMulZeroClass.toHasSmul.{u2, u3} 𝕜 E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} 𝕜 E (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u2, u3} 𝕜 E (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7))))) (f x) c) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) m f μ))
-but is expected to have type
- forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {𝕜 : Type.{u1}} [_inst_4 : NontriviallyNormedField.{u1} 𝕜] [_inst_5 : CompleteSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))))] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (Zero.toOfNat0.{u3} E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6))))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) m (fun (x : α) => HSMul.hSMul.{u1, u3, u3} 𝕜 E E (instHSMul.{u1, u3} 𝕜 E (SMulZeroClass.toSMul.{u1, u3} 𝕜 E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSMulZeroClass.{u1, u3} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u1, u3} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7)))))) (f x) c) μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α 𝕜 (UniformSpace.toTopologicalSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))))) m f μ))
+<too large>
Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_smul_const_iff aestronglyMeasurable_smul_const_iffₓ'. -/
theorem aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ :=
@@ -2667,10 +2655,7 @@ variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
/- warning: strongly_measurable.apply_continuous_linear_map -> StronglyMeasurable.apply_continuousLinearMap is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {m : MeasurableSpace.{u1} α} {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v))
-but is expected to have type
- forall {α : Type.{u4}} {𝕜 : Type.{u3}} [_inst_4 : NontriviallyNormedField.{u3} 𝕜] {E : Type.{u1}} [_inst_5 : NormedAddCommGroup.{u1} E] [_inst_6 : NormedSpace.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {m : MeasurableSpace.{u4} α} {φ : α -> (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u4, max u1 u2} α (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u3, u3, u2, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u4, u1} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)))) (φ a) v))
+<too large>
Case conversion may be inaccurate. Consider using '#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {φ : α → F →L[𝕜] E}
(hφ : StronglyMeasurable φ) (v : F) : StronglyMeasurable fun a => φ a v :=
@@ -2678,10 +2663,7 @@ theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
/- warning: measure_theory.ae_strongly_measurable.apply_continuous_linear_map -> MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v) μ)
-but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u4}} [_inst_4 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_5 : NormedAddCommGroup.{u2} E] [_inst_6 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u4, u4, u3, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)))) (φ a) v) μ)
+<too large>
Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
@@ -2689,10 +2671,7 @@ theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStrongly
#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
/- warning: continuous_linear_map.ae_strongly_measurable_comp₂ -> ContinuousLinearMap.aestronglyMeasurable_comp₂ is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {G : Type.{u5}} [_inst_9 : NormedAddCommGroup.{u5} G] [_inst_10 : NormedSpace.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)] (L : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) {f : α -> E} {g : α -> F}, (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u4} α F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u5} α G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) m (fun (x : α) => coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F 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_inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (coeFn.{max (succ u3) (succ (max u4 u5)), max (succ u3) (succ (max u4 u5))} (ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) (fun (_x : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) => E -> (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (ContinuousLinearMap.toFun.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) L (f x)) (g x)) μ)
-but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u5}} [_inst_4 : NontriviallyNormedField.{u5} 𝕜] {E : Type.{u4}} [_inst_5 : NormedAddCommGroup.{u4} E] [_inst_6 : NormedSpace.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {G : Type.{u3}} [_inst_9 : NormedAddCommGroup.{u3} G] [_inst_10 : NormedSpace.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)] (L : ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) 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_inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 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(AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) 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(PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) m (fun (x : α) => FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F 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(SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G 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u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) _x) (ContinuousMapClass.toFunLike.{max (max u4 u2) u3, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max (max u4 u2) u3, u5, u5, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u5, u4, max u2 u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))))) L (f x)) (g x)) μ)
+<too large>
Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂ₓ'. -/
theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946
@@ -3066,5 +3066,5 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
end MeasureTheory
-- Guard against import creep
-assert_not_exists inner_product_space
+assert_not_exists InnerProductSpace
mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
+! leanprover-community/mathlib commit ef95945cd48c932c9e034872bd25c3c220d9c946
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -17,6 +17,9 @@ import Mathbin.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly measurable functions
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
A function `f` is said to be strongly measurable if `f` is the sequential limit of simple functions.
It is said to be finitely strongly measurable with respect to a measure `μ` if the supports
of those simple functions have finite measure. We also provide almost everywhere versions of
@@ -678,33 +681,42 @@ end Arithmetic
section MulAction
-variable [TopologicalSpace β] {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable {M G G₀ : Type _}
+
+variable [TopologicalSpace β]
+
+variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
+
+variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
+
+variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
/- warning: strongly_measurable_const_smul_iff -> stronglyMeasurable_const_smul_iff is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G : Type.{u3}} [_inst_3 : Group.{u3} G] [_inst_4 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3))] [_inst_5 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3)) _inst_4)] {m : MeasurableSpace.{u1} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3)) _inst_4) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {G : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] [_inst_6 : Group.{u3} G] [_inst_7 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6))] [_inst_8 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_7)] {m : MeasurableSpace.{u1} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_7) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G : Type.{u1}} [_inst_3 : Group.{u1} G] [_inst_4 : MulAction.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))] [_inst_5 : ContinuousConstSMul.{u1, u2} G β _inst_2 (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)) _inst_4)] {m : MeasurableSpace.{u3} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} G β β (instHSMul.{u1, u2} G β (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)) _inst_4)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f)
+ forall {α : Type.{u3}} {β : Type.{u2}} {f : α -> β} [G : TopologicalSpace.{u2} β] {_inst_2 : Type.{u1}} [_inst_6 : Group.{u1} _inst_2] [_inst_7 : MulAction.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6))] [_inst_8 : ContinuousConstSMul.{u1, u2} _inst_2 β G (MulAction.toSMul.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6)) _inst_7)] {m : MeasurableSpace.{u3} α} (c : _inst_2), Iff (MeasureTheory.StronglyMeasurable.{u3, u2} α β G m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} _inst_2 β β (instHSMul.{u1, u2} _inst_2 β (MulAction.toSMul.{u1, u2} _inst_2 β (DivInvMonoid.toMonoid.{u1} _inst_2 (Group.toDivInvMonoid.{u1} _inst_2 _inst_6)) _inst_7)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β G m f)
Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iffₓ'. -/
theorem stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iff
-variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+theorem IsUnit.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} {c : M} (hc : IsUnit c) :
+ (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
+ let ⟨u, hu⟩ := hc
+ hu ▸ stronglyMeasurable_const_smul_iff u
+#align is_unit.strongly_measurable_const_smul_iff IsUnit.stronglyMeasurable_const_smul_iff
/- warning: strongly_measurable_const_smul_iff₀ -> stronglyMeasurable_const_smul_iff₀ is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G₀ : Type.{u3}} [_inst_6 : GroupWithZero.{u3} G₀] [_inst_7 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6))] [_inst_8 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)) _inst_7)] {m : MeasurableSpace.{u1} α} {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)))))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)) _inst_7) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {G₀ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] [_inst_9 : GroupWithZero.{u3} G₀] [_inst_10 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9))] [_inst_11 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)) _inst_10)] {m : MeasurableSpace.{u1} α} {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)))))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_9)) _inst_10) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] {G₀ : Type.{u2}} [_inst_6 : GroupWithZero.{u2} G₀] [_inst_7 : MulAction.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6))] [_inst_8 : ContinuousConstSMul.{u2, u1} G₀ β _inst_2 (MulAction.toSMul.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6)) _inst_7)] {m : MeasurableSpace.{u3} α} {c : G₀}, (Ne.{succ u2} G₀ c (OfNat.ofNat.{u2} G₀ 0 (Zero.toOfNat0.{u2} G₀ (MonoidWithZero.toZero.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u2, u1, u1} G₀ β β (instHSMul.{u2, u1} G₀ β (MulAction.toSMul.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6)) _inst_7)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 m f))
+ forall {α : Type.{u3}} {β : Type.{u1}} {f : α -> β} [G₀ : TopologicalSpace.{u1} β] {_inst_2 : Type.{u2}} [_inst_9 : GroupWithZero.{u2} _inst_2] [_inst_10 : MulAction.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9))] [_inst_11 : ContinuousConstSMul.{u2, u1} _inst_2 β G₀ (MulAction.toSMul.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9)) _inst_10)] {m : MeasurableSpace.{u3} α} {c : _inst_2}, (Ne.{succ u2} _inst_2 c (OfNat.ofNat.{u2} _inst_2 0 (Zero.toOfNat0.{u2} _inst_2 (MonoidWithZero.toZero.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α β G₀ m (fun (x : α) => HSMul.hSMul.{u2, u1, u1} _inst_2 β β (instHSMul.{u2, u1} _inst_2 β (MulAction.toSMul.{u2, u1} _inst_2 β (MonoidWithZero.toMonoid.{u2} _inst_2 (GroupWithZero.toMonoidWithZero.{u2} _inst_2 _inst_9)) _inst_10)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u1} α β G₀ m f))
Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀ₓ'. -/
theorem stronglyMeasurable_const_smul_iff₀ {m : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
- by
- refine' ⟨fun h => _, fun h => h.const_smul c⟩
- convert h.const_smul' c⁻¹
- simp [smul_smul, inv_mul_cancel hc]
+ (IsUnit.mk0 _ hc).stronglyMeasurable_const_smul_iff
#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀
end MulAction
@@ -2606,33 +2618,40 @@ end NormedSpace
section MulAction
-variable {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable {M G G₀ : Type _}
+
+variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
+
+variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
+
+variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
/- warning: ae_strongly_measurable_const_smul_iff -> aestronglyMeasurable_const_smul_iff is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u3}} [_inst_4 : Group.{u3} G] [_inst_5 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4))] [_inst_6 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4)) _inst_5)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4)) _inst_5) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u3}} [_inst_7 : Group.{u3} G] [_inst_8 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_7))] [_inst_9 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_7)) _inst_8)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_7)) _inst_8) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
but is expected to have type
- forall {α : Type.{u3}} {β : Type.{u2}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u1}} [_inst_4 : Group.{u1} G] [_inst_5 : MulAction.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4))] [_inst_6 : ContinuousConstSMul.{u1, u2} G β _inst_2 (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4)) _inst_5)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} G β β (instHSMul.{u1, u2} G β (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4)) _inst_5)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m f μ)
+ forall {α : Type.{u3}} {β : Type.{u2}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u1}} [_inst_7 : Group.{u1} G] [_inst_8 : MulAction.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_7))] [_inst_9 : ContinuousConstSMul.{u1, u2} G β _inst_2 (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_7)) _inst_8)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} G β β (instHSMul.{u1, u2} G β (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_7)) _inst_8)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m f μ)
Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iffₓ'. -/
theorem aestronglyMeasurable_const_smul_iff (c : G) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iff
-variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+theorem IsUnit.aEStronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
+ AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
+ let ⟨u, hu⟩ := hc
+ hu ▸ aestronglyMeasurable_const_smul_iff u
+#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aEStronglyMeasurable_const_smul_iff
/- warning: ae_strongly_measurable_const_smul_iff₀ -> aestronglyMeasurable_const_smul_iff₀ is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_7 : GroupWithZero.{u3} G₀] [_inst_8 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))] [_inst_9 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ))
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_10 : GroupWithZero.{u3} G₀] [_inst_11 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10))] [_inst_12 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10)) _inst_11)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10)))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10)) _inst_11) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ))
but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_7 : GroupWithZero.{u3} G₀] [_inst_8 : MulAction.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))] [_inst_9 : ContinuousConstSMul.{u3, u1} G₀ β _inst_2 (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (Zero.toOfNat0.{u3} G₀ (MonoidWithZero.toZero.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u3, u1, u1} G₀ β β (instHSMul.{u3, u1} G₀ β (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ))
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_10 : GroupWithZero.{u3} G₀] [_inst_11 : MulAction.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10))] [_inst_12 : ContinuousConstSMul.{u3, u1} G₀ β _inst_2 (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10)) _inst_11)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (Zero.toOfNat0.{u3} G₀ (MonoidWithZero.toZero.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u3, u1, u1} G₀ β β (instHSMul.{u3, u1} G₀ β (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_10)) _inst_11)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ))
Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀ₓ'. -/
theorem aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
- by
- refine' ⟨fun h => _, fun h => h.const_smul c⟩
- convert h.const_smul' c⁻¹
- simp [smul_smul, inv_mul_cancel hc]
+ (IsUnit.mk0 _ hc).aestronglyMeasurable_const_smul_iff
#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀
end MulAction
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -2649,9 +2649,9 @@ variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
/- warning: strongly_measurable.apply_continuous_linear_map -> StronglyMeasurable.apply_continuousLinearMap is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {m : MeasurableSpace.{u1} α} {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v))
+ forall {α : Type.{u1}} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {m : MeasurableSpace.{u1} α} {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v))
but is expected to have type
- forall {α : Type.{u4}} {𝕜 : Type.{u3}} [_inst_4 : NontriviallyNormedField.{u3} 𝕜] {E : Type.{u1}} [_inst_5 : NormedAddCommGroup.{u1} E] [_inst_6 : NormedSpace.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {m : MeasurableSpace.{u4} α} {φ : α -> (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u4, max u1 u2} α (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u3, u3, u2, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u4, u1} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)))) (φ a) v))
+ forall {α : Type.{u4}} {𝕜 : Type.{u3}} [_inst_4 : NontriviallyNormedField.{u3} 𝕜] {E : Type.{u1}} [_inst_5 : NormedAddCommGroup.{u1} E] [_inst_6 : NormedSpace.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {m : MeasurableSpace.{u4} α} {φ : α -> (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u4, max u1 u2} α (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u3, u3, u2, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u4, u1} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)))) (φ a) v))
Case conversion may be inaccurate. Consider using '#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {φ : α → F →L[𝕜] E}
(hφ : StronglyMeasurable φ) (v : F) : StronglyMeasurable fun a => φ a v :=
@@ -2660,9 +2660,9 @@ theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {
/- warning: measure_theory.ae_strongly_measurable.apply_continuous_linear_map -> MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v) μ)
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v) μ)
but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u4}} [_inst_4 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_5 : NormedAddCommGroup.{u2} E] [_inst_6 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u4, u4, u3, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)))) (φ a) v) μ)
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u4}} [_inst_4 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_5 : NormedAddCommGroup.{u2} E] [_inst_6 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u4, u4, u3, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)))) (φ a) v) μ)
Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
@@ -2671,9 +2671,9 @@ theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStrongly
/- warning: continuous_linear_map.ae_strongly_measurable_comp₂ -> ContinuousLinearMap.aestronglyMeasurable_comp₂ is a dubious translation:
lean 3 declaration is
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {G : Type.{u5}} [_inst_9 : NormedAddCommGroup.{u5} G] [_inst_10 : NormedSpace.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)] (L : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) {f : α -> E} {g : α -> F}, (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u4} α F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u5} α G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) m (fun (x : α) => coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) => F -> G) (ContinuousLinearMap.toFun.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 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(ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F 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(NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) (fun (_x : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) => E -> (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (ContinuousLinearMap.toFun.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) L (f x)) (g x)) μ)
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {G : Type.{u5}} [_inst_9 : NormedAddCommGroup.{u5} G] [_inst_10 : NormedSpace.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)] (L : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} 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(NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) {f : α -> E} {g : α -> F}, (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u4} α F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u5} α G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) m (fun (x : α) => coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) => F -> G) (ContinuousLinearMap.toFun.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 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_inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (coeFn.{max (succ u3) (succ (max u4 u5)), max (succ u3) (succ (max u4 u5))} (ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 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(NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) (fun (_x : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) => E -> (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (ContinuousLinearMap.toFun.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) L (f x)) (g x)) μ)
but is expected to have type
- forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u5}} [_inst_4 : NontriviallyNormedField.{u5} 𝕜] {E : Type.{u4}} [_inst_5 : NormedAddCommGroup.{u4} E] [_inst_6 : NormedSpace.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {G : Type.{u3}} [_inst_9 : NormedAddCommGroup.{u3} G] [_inst_10 : NormedSpace.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)] (L : ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) 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(ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G 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(NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) 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(Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) _x) (ContinuousMapClass.toFunLike.{max (max u4 u2) u3, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max (max u4 u2) u3, u5, u5, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u5, u4, max u2 u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))))) L (f x)) (g x)) μ)
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u5}} [_inst_4 : NontriviallyNormedField.{u5} 𝕜] {E : Type.{u4}} [_inst_5 : NormedAddCommGroup.{u4} E] [_inst_6 : NormedSpace.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {G : Type.{u3}} [_inst_9 : NormedAddCommGroup.{u3} G] [_inst_10 : NormedSpace.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)] (L : ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G 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(SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) _x) (ContinuousMapClass.toFunLike.{max (max u4 u2) u3, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max (max u4 u2) u3, u5, u5, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u5, u4, max u2 u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.toTopologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))))) L (f x)) (g x)) μ)
Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂ₓ'. -/
theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -67,6 +67,7 @@ open MeasureTheory Filter TopologicalSpace Function Set MeasureTheory.Measure
open ENNReal Topology MeasureTheory NNReal BigOperators
+#print SecondCountableTopologyEither /-
/-- The typeclass `second_countable_topology_either α β` registers the fact that at least one of
the two spaces has second countable topology. This is the right assumption to ensure that continuous
maps from `α` to `β` are strongly measurable. -/
@@ -74,16 +75,21 @@ class SecondCountableTopologyEither (α β : Type _) [TopologicalSpace α] [Topo
Prop where
out : SecondCountableTopology α ∨ SecondCountableTopology β
#align second_countable_topology_either SecondCountableTopologyEither
+-/
+#print secondCountableTopologyEither_of_left /-
instance (priority := 100) secondCountableTopologyEither_of_left (α β : Type _) [TopologicalSpace α]
[TopologicalSpace β] [SecondCountableTopology α] : SecondCountableTopologyEither α β
where out := Or.inl (by infer_instance)
#align second_countable_topology_either_of_left secondCountableTopologyEither_of_left
+-/
+#print secondCountableTopologyEither_of_right /-
instance (priority := 100) secondCountableTopologyEither_of_right (α β : Type _)
[TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] :
SecondCountableTopologyEither α β where out := Or.inr (by infer_instance)
#align second_countable_topology_either_of_right secondCountableTopologyEither_of_right
+-/
variable {α β γ ι : Type _} [Countable ι]
@@ -96,31 +102,39 @@ section Definitions
variable [TopologicalSpace β]
+#print MeasureTheory.StronglyMeasurable /-
/-- A function is `strongly_measurable` if it is the limit of simple functions. -/
def StronglyMeasurable [MeasurableSpace α] (f : α → β) : Prop :=
∃ fs : ℕ → α →ₛ β, ∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
#align measure_theory.strongly_measurable MeasureTheory.StronglyMeasurable
+-/
-- mathport name: strongly_measurable_of
scoped notation "strongly_measurable[" m "]" => @MeasureTheory.StronglyMeasurable _ _ _ m
+#print MeasureTheory.FinStronglyMeasurable /-
/-- A function is `fin_strongly_measurable` with respect to a measure if it is the limit of simple
functions with support with finite measure. -/
def FinStronglyMeasurable [Zero β] {m0 : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ fs : ℕ → α →ₛ β, (∀ n, μ (support (fs n)) < ∞) ∧ ∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
#align measure_theory.fin_strongly_measurable MeasureTheory.FinStronglyMeasurable
+-/
+#print MeasureTheory.AEStronglyMeasurable /-
/-- A function is `ae_strongly_measurable` with respect to a measure `μ` if it is almost everywhere
equal to the limit of a sequence of simple functions. -/
-def AeStronglyMeasurable {m0 : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
+def AEStronglyMeasurable {m0 : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g, StronglyMeasurable g ∧ f =ᵐ[μ] g
-#align measure_theory.ae_strongly_measurable MeasureTheory.AeStronglyMeasurable
+#align measure_theory.ae_strongly_measurable MeasureTheory.AEStronglyMeasurable
+-/
+#print MeasureTheory.AEFinStronglyMeasurable /-
/-- A function is `ae_fin_strongly_measurable` with respect to a measure if it is almost everywhere
equal to the limit of a sequence of simple functions with support with finite measure. -/
-def AeFinStronglyMeasurable [Zero β] {m0 : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
+def AEFinStronglyMeasurable [Zero β] {m0 : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g, FinStronglyMeasurable g μ ∧ f =ᵐ[μ] g
-#align measure_theory.ae_fin_strongly_measurable MeasureTheory.AeFinStronglyMeasurable
+#align measure_theory.ae_fin_strongly_measurable MeasureTheory.AEFinStronglyMeasurable
+-/
end Definitions
@@ -129,12 +143,24 @@ open MeasureTheory
/-! ## Strongly measurable functions -/
-protected theorem StronglyMeasurable.aeStronglyMeasurable {α β} {m0 : MeasurableSpace α}
+/- warning: measure_theory.strongly_measurable.ae_strongly_measurable -> MeasureTheory.StronglyMeasurable.aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {μ : MeasureTheory.Measure.{u1} α m0}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m0 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m0 f μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m0 : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {μ : MeasureTheory.Measure.{u2} α m0}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m0 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m0 f μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurableₓ'. -/
+protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {m0 : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
- AeStronglyMeasurable f μ :=
+ AEStronglyMeasurable f μ :=
⟨f, hf, EventuallyEq.refl _ _⟩
-#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aeStronglyMeasurable
-
+#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurable
+
+/- warning: measure_theory.subsingleton.strongly_measurable -> MeasureTheory.Subsingleton.stronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : Subsingleton.{succ u2} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : Subsingleton.{succ u1} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f
+Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurableₓ'. -/
@[simp]
theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton β] (f : α → β) : StronglyMeasurable f :=
@@ -148,21 +174,45 @@ theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [Topologica
exact MeasurableSet.univ
#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable
+/- warning: measure_theory.simple_func.strongly_measurable -> MeasureTheory.SimpleFunc.stronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α m β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α m β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurableₓ'. -/
theorem SimpleFunc.stronglyMeasurable {α β} {m : MeasurableSpace α} [TopologicalSpace β]
(f : α →ₛ β) : StronglyMeasurable f :=
⟨fun _ => f, fun x => tendsto_const_nhds⟩
#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable
+/- warning: measure_theory.strongly_measurable_of_is_empty -> MeasureTheory.stronglyMeasurable_of_isEmpty is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : IsEmpty.{succ u1} α] {m : MeasurableSpace.{u1} α} [_inst_3 : TopologicalSpace.{u2} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 m f
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : IsEmpty.{succ u2} α] {m : MeasurableSpace.{u2} α} [_inst_3 : TopologicalSpace.{u1} β] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 m f
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmptyₓ'. -/
theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {m : MeasurableSpace α} [TopologicalSpace β]
(f : α → β) : StronglyMeasurable f :=
⟨fun n => SimpleFunc.ofIsEmpty, isEmptyElim⟩
#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
+/- warning: measure_theory.strongly_measurable_const -> MeasureTheory.stronglyMeasurable_const is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {b : β}, MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (a : α) => b)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {b : β}, MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (fun (a : α) => b)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_constₓ'. -/
theorem stronglyMeasurable_const {α β} {m : MeasurableSpace α} [TopologicalSpace β] {b : β} :
StronglyMeasurable fun a : α => b :=
⟨fun n => SimpleFunc.const α b, fun a => tendsto_const_nhds⟩
#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_const
+/- warning: measure_theory.strongly_measurable_one -> MeasureTheory.stronglyMeasurable_one is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : One.{u2} β], MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 1 (OfNat.mk.{max u1 u2} (α -> β) 1 (One.one.{max u1 u2} (α -> β) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : One.{u1} β], MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 1 (One.toOfNat1.{max u2 u1} (α -> β) (Pi.instOne.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.1583 : α) => β) (fun (i : α) => _inst_3))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_oneₓ'. -/
@[to_additive]
theorem stronglyMeasurable_one {α β} {m : MeasurableSpace α} [TopologicalSpace β] [One β] :
StronglyMeasurable (1 : α → β) :=
@@ -170,6 +220,12 @@ theorem stronglyMeasurable_one {α β} {m : MeasurableSpace α} [TopologicalSpac
#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_one
#align measure_theory.strongly_measurable_zero MeasureTheory.stronglyMeasurable_zero
+/- warning: measure_theory.strongly_measurable_const' -> MeasureTheory.stronglyMeasurable_const' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (forall (x : α) (y : α), Eq.{succ u2} β (f x) (f y)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (forall (x : α) (y : α), Eq.{succ u1} β (f x) (f y)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'ₓ'. -/
/-- A version of `strongly_measurable_const` that assumes `f x = f y` for all `x, y`.
This version works for functions between empty types. -/
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
@@ -181,11 +237,17 @@ theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalS
exact funext fun x => hf x h.some
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
+/- warning: measure_theory.subsingleton.strongly_measurable' -> MeasureTheory.Subsingleton.stronglyMeasurable' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : Subsingleton.{succ u1} α] (f : α -> β), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : Subsingleton.{succ u2} α] (f : α -> β), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f
+Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.stronglyMeasurable'ₓ'. -/
@[simp]
-theorem Subsingleton.strongly_measurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
+theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton α] (f : α → β) : StronglyMeasurable f :=
stronglyMeasurable_const' fun x y => by rw [Subsingleton.elim x y]
-#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.strongly_measurable'
+#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.stronglyMeasurable'
namespace StronglyMeasurable
@@ -195,18 +257,27 @@ section BasicPropertiesInAnyTopologicalSpace
variable [TopologicalSpace β]
+#print MeasureTheory.StronglyMeasurable.approx /-
/-- A sequence of simple functions such that `∀ x, tendsto (λ n, hf.approx n x) at_top (𝓝 (f x))`.
That property is given by `strongly_measurable.tendsto_approx`. -/
protected noncomputable def approx {m : MeasurableSpace α} (hf : StronglyMeasurable f) :
ℕ → α →ₛ β :=
hf.some
#align measure_theory.strongly_measurable.approx MeasureTheory.StronglyMeasurable.approx
+-/
+/- warning: measure_theory.strongly_measurable.tendsto_approx -> MeasureTheory.StronglyMeasurable.tendsto_approx is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (x : α), Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approx.{u1, u2} α β f _inst_2 m hf n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β _inst_2 (f x))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] {m : MeasurableSpace.{u2} α} (hf : MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (x : α), Filter.Tendsto.{0, u1} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β (MeasureTheory.StronglyMeasurable.approx.{u2, u1} α β f _inst_2 m hf n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} β _inst_2 (f x))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx MeasureTheory.StronglyMeasurable.tendsto_approxₓ'. -/
protected theorem tendsto_approx {m : MeasurableSpace α} (hf : StronglyMeasurable f) :
∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
hf.choose_spec
#align measure_theory.strongly_measurable.tendsto_approx MeasureTheory.StronglyMeasurable.tendsto_approx
+#print MeasureTheory.StronglyMeasurable.approxBounded /-
/-- Similar to `strongly_measurable.approx`, but enforces that the norm of every function in the
sequence is less than `c` everywhere. If `‖f x‖ ≤ c` this sequence of simple functions verifies
`tendsto (λ n, hf.approx_bounded n x) at_top (𝓝 (f x))`. -/
@@ -214,7 +285,14 @@ noncomputable def approxBounded {m : MeasurableSpace α} [Norm β] [SMul ℝ β]
(hf : StronglyMeasurable f) (c : ℝ) : ℕ → SimpleFunc α β := fun n =>
(hf.approx n).map fun x => min 1 (c / ‖x‖) • x
#align measure_theory.strongly_measurable.approx_bounded MeasureTheory.StronglyMeasurable.approxBounded
+-/
+/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real} {x : α}, (LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (f x)) c) -> (Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3))) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x)))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real} {x : α}, (LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (NormedAddCommGroup.toNorm.{u2} β _inst_3) (f x)) c) -> (Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_leₓ'. -/
theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} (hf : strongly_measurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) :=
@@ -252,6 +330,12 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
refine' tendsto.div tendsto_const_nhds h_tendsto.norm hfx0
#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
+/- warning: measure_theory.strongly_measurable.tendsto_approx_bounded_ae -> MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real}, (Filter.Eventually.{u1} α (fun (x : α) => LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (f x)) c) (MeasureTheory.Measure.ae.{u1} α m0 μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3))) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x))) (MeasureTheory.Measure.ae.{u1} α m0 μ))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : NormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m f) {c : Real}, (Filter.Eventually.{u1} α (fun (x : α) => LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (NormedAddCommGroup.toNorm.{u2} β _inst_3) (f x)) c) (MeasureTheory.Measure.ae.{u1} α m0 μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) m (NormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (NormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3) _inst_4))))) hf c n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} β _inst_3)))) (f x))) (MeasureTheory.Measure.ae.{u1} α m0 μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_aeₓ'. -/
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : strongly_measurable[m] f) {c : ℝ}
(hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
@@ -259,6 +343,12 @@ theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [N
filter_upwards [hf_bound]with x hfx using tendsto_approx_bounded_of_norm_le hf hfx
#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
+/- warning: measure_theory.strongly_measurable.norm_approx_bounded_le -> MeasureTheory.StronglyMeasurable.norm_approxBounded_le is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : SeminormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField _inst_3] {m : MeasurableSpace.{u1} α} {c : Real} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m f), (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) c) -> (forall (n : Nat) (x : α), LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_3) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_3) (SMulZeroClass.toHasSmul.{0, u2} Real β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real β (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3))))) (Module.toMulActionWithZero.{0, u2} Real β (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField _inst_3 _inst_4))))) hf c n) x)) c)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_3 : SeminormedAddCommGroup.{u2} β] [_inst_4 : NormedSpace.{0, u2} Real β Real.normedField _inst_3] {m : MeasurableSpace.{u1} α} {c : Real} (hf : MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m f), (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) c) -> (forall (n : Nat) (x : α), LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (SeminormedAddCommGroup.toNorm.{u2} β _inst_3) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α m β (MeasureTheory.StronglyMeasurable.approxBounded.{u1, u2} α β f (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_3))) m (SeminormedAddCommGroup.toNorm.{u2} β _inst_3) (SMulZeroClass.toSMul.{0, u2} Real β (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real β Real.instZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real β Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (SubtractionCommMonoid.toSubtractionMonoid.{u2} β (AddCommGroup.toDivisionAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)))))) (Module.toMulActionWithZero.{0, u2} Real β Real.semiring (AddCommGroup.toAddCommMonoid.{u2} β (SeminormedAddCommGroup.toAddCommGroup.{u2} β _inst_3)) (NormedSpace.toModule.{0, u2} Real β Real.normedField _inst_3 _inst_4))))) hf c n) x)) c)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_leₓ'. -/
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : strongly_measurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
‖hf.approxBounded c n x‖ ≤ c :=
@@ -279,6 +369,12 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
· rwa [div_le_one (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]
#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_le
+/- warning: strongly_measurable_bot_iff -> stronglyMeasurable_bot_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Nonempty.{succ u2} β] [_inst_4 : T2Space.{u2} β _inst_2], Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 (Bot.bot.{u1} (MeasurableSpace.{u1} α) (CompleteLattice.toHasBot.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.completeLattice.{u1} α))) f) (Exists.{succ u2} β (fun (c : β) => Eq.{max (succ u1) (succ u2)} (α -> β) f (fun (_x : α) => c)))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Nonempty.{succ u2} β] [_inst_4 : T2Space.{u2} β _inst_2], Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 (Bot.bot.{u1} (MeasurableSpace.{u1} α) (CompleteLattice.toBot.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u1} α))) f) (Exists.{succ u2} β (fun (c : β) => Eq.{max (succ u1) (succ u2)} (α -> β) f (fun (_x : α) => c)))
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_bot_iff stronglyMeasurable_bot_iffₓ'. -/
theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
strongly_measurable[⊥] f ↔ ∃ c, f = fun _ => c :=
by
@@ -301,6 +397,12 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
+/- warning: measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite -> MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {t : Set.{u1} α}, (MeasurableSet.{u1} α m t) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_3))))) -> (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t)) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {t : Set.{u1} α}, (MeasurableSet.{u1} α m t) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_3)))) -> (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t)) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFiniteₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
@@ -355,6 +457,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
exact hn₂ m ((le_max_right _ _).trans hm.le)
#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
+#print MeasureTheory.StronglyMeasurable.finStronglyMeasurable /-
/-- If the measure is sigma-finite, all strongly measurable functions are
`fin_strongly_measurable`. -/
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
@@ -362,7 +465,14 @@ protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : Me
hf.finStronglyMeasurable_of_set_sigmaFinite MeasurableSet.univ (by simp)
(by rwa [measure.restrict_univ])
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
+-/
+/- warning: measure_theory.strongly_measurable.measurable -> MeasureTheory.StronglyMeasurable.measurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (Measurable.{u1, u2} α β m _inst_4 f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (Measurable.{u2, u1} α β m _inst_4 f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurableₓ'. -/
/-- A strongly measurable function is measurable. -/
protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f :=
@@ -370,19 +480,37 @@ protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [Pse
(tendsto_pi_nhds.mpr hf.tendsto_approx)
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
+/- warning: measure_theory.strongly_measurable.ae_measurable -> MeasureTheory.StronglyMeasurable.aemeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4] {μ : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (AEMeasurable.{u1, u2} α β _inst_4 m f μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4] {μ : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (AEMeasurable.{u2, u1} α β _inst_4 m f μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurableₓ'. -/
/-- A strongly measurable function is almost everywhere measurable. -/
-protected theorem aEMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
+protected theorem aemeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
(hf : StronglyMeasurable f) : AEMeasurable f μ :=
hf.Measurable.AEMeasurable
-#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aEMeasurable
-
+#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurable
+
+/- warning: continuous.comp_strongly_measurable -> Continuous.comp_stronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u1} β γ _inst_2 _inst_3 g) -> (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u3, u1} α γ _inst_3 m (fun (x : α) => g (f x)))
+Case conversion may be inaccurate. Consider using '#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurableₓ'. -/
theorem Continuous.comp_stronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => g (f x) :=
⟨fun n => SimpleFunc.map g (hf.approx n), fun x => (hg.Tendsto _).comp (hf.tendsto_approx x)⟩
#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurable
+/- warning: measure_theory.strongly_measurable.measurable_set_mul_support -> MeasureTheory.StronglyMeasurable.measurableSet_mulSupport is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : One.{u2} β] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : TopologicalSpace.MetrizableSpace.{u2} β _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_3 m f) -> (MeasurableSet.{u1} α m (Function.mulSupport.{u1, u2} α β _inst_2 f))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : One.{u1} β] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : TopologicalSpace.MetrizableSpace.{u1} β _inst_3], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_3 m f) -> (MeasurableSet.{u2} α m (Function.mulSupport.{u2, u1} α β _inst_2 f))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupportₓ'. -/
@[to_additive]
theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
[MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) :=
@@ -392,6 +520,12 @@ theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalS
#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
#align measure_theory.strongly_measurable.measurable_set_support MeasureTheory.StronglyMeasurable.measurableSet_support
+/- warning: measure_theory.strongly_measurable.mono -> MeasureTheory.StronglyMeasurable.mono is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} {m' : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m' f) -> (LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m' m) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} {m' : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m' f) -> (LE.le.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instLEMeasurableSpace.{u2} α) m' m) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.monoₓ'. -/
protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
(hf : strongly_measurable[m'] f) (h_mono : m' ≤ m) : strongly_measurable[m] f :=
by
@@ -402,6 +536,12 @@ protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
exact ⟨f_approx, hf.tendsto_approx⟩
#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.mono
+/- warning: measure_theory.strongly_measurable.prod_mk -> MeasureTheory.StronglyMeasurable.prod_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_3 m g) -> (MeasureTheory.StronglyMeasurable.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u3} β γ (f x) (g x)))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u3, u1} α γ _inst_3 m g) -> (MeasureTheory.StronglyMeasurable.{u3, max u1 u2} α (Prod.{u2, u1} β γ) (instTopologicalSpaceProd.{u2, u1} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u1} β γ (f x) (g x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mkₓ'. -/
protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ]
{f : α → β} {g : α → γ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => (f x, g x) :=
@@ -411,17 +551,35 @@ protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [Topolo
exact tendsto.prod_mk (hf.tendsto_approx x) (hg.tendsto_approx x)
#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mk
+/- warning: measure_theory.strongly_measurable.comp_measurable -> MeasureTheory.StronglyMeasurable.comp_measurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {m : MeasurableSpace.{u1} α} {m' : MeasurableSpace.{u3} γ} {f : α -> β} {g : γ -> α}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (Measurable.{u3, u1} γ α m' m g) -> (MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 m' (Function.comp.{succ u3, succ u1, succ u2} γ α β f g))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {m : MeasurableSpace.{u2} α} {m' : MeasurableSpace.{u1} γ} {f : α -> β} {g : γ -> α}, (MeasureTheory.StronglyMeasurable.{u2, u3} α β _inst_2 m f) -> (Measurable.{u1, u2} γ α m' m g) -> (MeasureTheory.StronglyMeasurable.{u1, u3} γ β _inst_2 m' (Function.comp.{succ u1, succ u2, succ u3} γ α β f g))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurableₓ'. -/
theorem comp_measurable [TopologicalSpace β] {m : MeasurableSpace α} {m' : MeasurableSpace γ}
{f : α → β} {g : γ → α} (hf : StronglyMeasurable f) (hg : Measurable g) :
StronglyMeasurable (f ∘ g) :=
⟨fun n => SimpleFunc.comp (hf.approx n) g hg, fun x => hf.tendsto_approx (g x)⟩
#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurable
+/- warning: measure_theory.strongly_measurable.of_uncurry_left -> MeasureTheory.StronglyMeasurable.of_uncurry_left is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u3, u2} (Prod.{u1, u3} α γ) β _inst_2 (Prod.instMeasurableSpace.{u1, u3} α γ mα mγ) (Function.uncurry.{u1, u3, u2} α γ β f)) -> (forall {x : α}, MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 mγ (f x))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {mα : MeasurableSpace.{u2} α} {mγ : MeasurableSpace.{u1} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u2, u3} (Prod.{u2, u1} α γ) β _inst_2 (Prod.instMeasurableSpace.{u2, u1} α γ mα mγ) (Function.uncurry.{u2, u1, u3} α γ β f)) -> (forall {x : α}, MeasureTheory.StronglyMeasurable.{u1, u3} γ β _inst_2 mγ (f x))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_leftₓ'. -/
theorem of_uncurry_left [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) :=
hf.comp_measurable measurable_prod_mk_left
#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_left
+/- warning: measure_theory.strongly_measurable.of_uncurry_right -> MeasureTheory.StronglyMeasurable.of_uncurry_right is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u3, u2} (Prod.{u1, u3} α γ) β _inst_2 (Prod.instMeasurableSpace.{u1, u3} α γ mα mγ) (Function.uncurry.{u1, u3, u2} α γ β f)) -> (forall {y : γ}, MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (fun (x : α) => f x y))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {mα : MeasurableSpace.{u2} α} {mγ : MeasurableSpace.{u1} γ} {f : α -> γ -> β}, (MeasureTheory.StronglyMeasurable.{max u1 u2, u3} (Prod.{u2, u1} α γ) β _inst_2 (Prod.instMeasurableSpace.{u2, u1} α γ mα mγ) (Function.uncurry.{u2, u1, u3} α γ β f)) -> (forall {y : γ}, MeasureTheory.StronglyMeasurable.{u2, u3} α β _inst_2 mα (fun (x : α) => f x y))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.of_uncurry_right MeasureTheory.StronglyMeasurable.of_uncurry_rightₓ'. -/
theorem of_uncurry_right [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
StronglyMeasurable fun x => f x y :=
@@ -434,27 +592,39 @@ variable {mα : MeasurableSpace α} [TopologicalSpace β]
include mα
+#print MeasureTheory.StronglyMeasurable.mul /-
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f * g) :=
⟨fun n => hf.approx n * hg.approx n, fun x => (hf.tendsto_approx x).mul (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.mul MeasureTheory.StronglyMeasurable.mul
#align measure_theory.strongly_measurable.add MeasureTheory.StronglyMeasurable.add
+-/
+#print MeasureTheory.StronglyMeasurable.mul_const /-
@[to_additive]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
StronglyMeasurable fun x => f x * c :=
hf.mul stronglyMeasurable_const
#align measure_theory.strongly_measurable.mul_const MeasureTheory.StronglyMeasurable.mul_const
#align measure_theory.strongly_measurable.add_const MeasureTheory.StronglyMeasurable.add_const
+-/
+#print MeasureTheory.StronglyMeasurable.const_mul /-
@[to_additive]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
StronglyMeasurable fun x => c * f x :=
stronglyMeasurable_const.mul hf
#align measure_theory.strongly_measurable.const_mul MeasureTheory.StronglyMeasurable.const_mul
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
+-/
+/- warning: measure_theory.strongly_measurable.inv -> MeasureTheory.StronglyMeasurable.inv is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {mα : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Group.{u2} β] [_inst_4 : TopologicalGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (Inv.inv.{max u1 u2} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasInv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_3))) f))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {mα : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Group.{u2} β] [_inst_4 : TopologicalGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα (Inv.inv.{max u2 u1} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => InvOneClass.toInv.{u2} β (DivInvOneMonoid.toInvOneClass.{u2} β (DivisionMonoid.toDivInvOneMonoid.{u2} β (Group.toDivisionMonoid.{u2} β _inst_3))))) f))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.invₓ'. -/
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
@@ -462,13 +632,16 @@ protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable
#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv
#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg
+#print MeasureTheory.StronglyMeasurable.div /-
@[to_additive]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f / g) :=
⟨fun n => hf.approx n / hg.approx n, fun x => (hf.tendsto_approx x).div' (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.div MeasureTheory.StronglyMeasurable.div
#align measure_theory.strongly_measurable.sub MeasureTheory.StronglyMeasurable.sub
+-/
+#print MeasureTheory.StronglyMeasurable.smul /-
@[to_additive]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
@@ -476,23 +649,30 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
continuous_smul.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.smul MeasureTheory.StronglyMeasurable.smul
#align measure_theory.strongly_measurable.vadd MeasureTheory.StronglyMeasurable.vadd
+-/
+#print MeasureTheory.StronglyMeasurable.const_smul /-
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable (c • f) :=
⟨fun n => c • hf.approx n, fun x => (hf.tendsto_approx x).const_smul c⟩
#align measure_theory.strongly_measurable.const_smul MeasureTheory.StronglyMeasurable.const_smul
+-/
+#print MeasureTheory.StronglyMeasurable.const_smul' /-
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable fun x => c • f x :=
hf.const_smul c
#align measure_theory.strongly_measurable.const_smul' MeasureTheory.StronglyMeasurable.const_smul'
+-/
+#print MeasureTheory.StronglyMeasurable.smul_const /-
@[to_additive]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
(hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c :=
continuous_smul.comp_stronglyMeasurable (hf.prod_mk stronglyMeasurable_const)
#align measure_theory.strongly_measurable.smul_const MeasureTheory.StronglyMeasurable.smul_const
#align measure_theory.strongly_measurable.vadd_const MeasureTheory.StronglyMeasurable.vadd_const
+-/
end Arithmetic
@@ -500,6 +680,12 @@ section MulAction
variable [TopologicalSpace β] {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
+/- warning: strongly_measurable_const_smul_iff -> stronglyMeasurable_const_smul_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G : Type.{u3}} [_inst_3 : Group.{u3} G] [_inst_4 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3))] [_inst_5 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3)) _inst_4)] {m : MeasurableSpace.{u1} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_3)) _inst_4) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G : Type.{u1}} [_inst_3 : Group.{u1} G] [_inst_4 : MulAction.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))] [_inst_5 : ContinuousConstSMul.{u1, u2} G β _inst_2 (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)) _inst_4)] {m : MeasurableSpace.{u3} α} (c : G), Iff (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} G β β (instHSMul.{u1, u2} G β (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)) _inst_4)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f)
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iffₓ'. -/
theorem stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
@@ -507,6 +693,12 @@ theorem stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+/- warning: strongly_measurable_const_smul_iff₀ -> stronglyMeasurable_const_smul_iff₀ is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {G₀ : Type.{u3}} [_inst_6 : GroupWithZero.{u3} G₀] [_inst_7 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6))] [_inst_8 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)) _inst_7)] {m : MeasurableSpace.{u1} α} {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)))))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_6)) _inst_7) c (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] {G₀ : Type.{u2}} [_inst_6 : GroupWithZero.{u2} G₀] [_inst_7 : MulAction.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6))] [_inst_8 : ContinuousConstSMul.{u2, u1} G₀ β _inst_2 (MulAction.toSMul.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6)) _inst_7)] {m : MeasurableSpace.{u3} α} {c : G₀}, (Ne.{succ u2} G₀ c (OfNat.ofNat.{u2} G₀ 0 (Zero.toOfNat0.{u2} G₀ (MonoidWithZero.toZero.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6))))) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u2, u1, u1} G₀ β β (instHSMul.{u2, u1} G₀ β (MulAction.toSMul.{u2, u1} G₀ β (MonoidWithZero.toMonoid.{u2} G₀ (GroupWithZero.toMonoidWithZero.{u2} G₀ _inst_6)) _inst_7)) c (f x))) (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 m f))
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀ₓ'. -/
theorem stronglyMeasurable_const_smul_iff₀ {m : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
by
@@ -525,17 +717,21 @@ open Filter
open Filter
+#print MeasureTheory.StronglyMeasurable.sup /-
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
(hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
+-/
+#print MeasureTheory.StronglyMeasurable.inf /-
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
(hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.inf MeasureTheory.StronglyMeasurable.inf
+-/
end Order
@@ -550,6 +746,12 @@ variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : Mea
include m
+/- warning: list.strongly_measurable_prod' -> List.stronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : Monoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))] {m : MeasurableSpace.{u1} α} (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (List.prod.{max u1 u2} (α -> M) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))) l))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : Monoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))] {m : MeasurableSpace.{u2} α} (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (List.prod.{max u2 u1} (α -> M) (Pi.instMul.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))) (Pi.instOne.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => Monoid.toOne.{u1} M _inst_2)) l))
+Case conversion may be inaccurate. Consider using '#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable l.Prod := by
@@ -560,6 +762,12 @@ theorem List.stronglyMeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, S
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
#align list.strongly_measurable_sum' List.stronglyMeasurable_sum'
+/- warning: list.strongly_measurable_prod -> List.stronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : Monoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2))] {m : MeasurableSpace.{u1} α} (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (x : α) => List.prod.{u2} M (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2)) (MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_2)) (List.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) l)))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : Monoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2))] {m : MeasurableSpace.{u2} α} (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (x : α) => List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (Monoid.toOne.{u1} M _inst_2) (List.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) l)))
+Case conversion may be inaccurate. Consider using '#align list.strongly_measurable_prod List.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem List.stronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable fun x => (l.map fun f : α → M => f x).Prod := by
@@ -575,6 +783,12 @@ variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m :
include m
+/- warning: multiset.strongly_measurable_prod' -> Multiset.stronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} (l : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (Multiset.prod.{max u1 u2} (α -> M) (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) l))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} (l : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (Multiset.prod.{max u2 u1} (α -> M) (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) l))
+Case conversion may be inaccurate. Consider using '#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.Prod :=
@@ -584,6 +798,12 @@ theorem Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'
#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'
+/- warning: multiset.strongly_measurable_prod -> Multiset.stronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} (s : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (x : α) => Multiset.prod.{u2} M _inst_2 (Multiset.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) s)))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} (s : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (x : α) => Multiset.prod.{u1} M _inst_2 (Multiset.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) s)))
+Case conversion may be inaccurate. Consider using '#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, StronglyMeasurable f) :
@@ -592,6 +812,12 @@ theorem Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prod
#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum
+/- warning: finset.strongly_measurable_prod' -> Finset.stronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) s (fun (i : ι) => f i)))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_2)) s (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'ₓ'. -/
@[to_additive]
theorem Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i in s, f i) :=
@@ -599,6 +825,12 @@ theorem Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s :
#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'
#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'
+/- warning: finset.strongly_measurable_prod -> Finset.stronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {M : Type.{u2}} [_inst_2 : CommMonoid.{u2} M] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : ContinuousMul.{u2} M _inst_3 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_2)))] {m : MeasurableSpace.{u1} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α M _inst_3 m (fun (a : α) => Finset.prod.{u2, u3} M ι _inst_2 s (fun (i : ι) => f i a)))
+but is expected to have type
+ forall {α : Type.{u2}} {M : Type.{u1}} [_inst_2 : CommMonoid.{u1} M] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : ContinuousMul.{u1} M _inst_3 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_2)))] {m : MeasurableSpace.{u2} α} {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (f i))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α M _inst_3 m (fun (a : α) => Finset.prod.{u1, u3} M ι _inst_2 s (fun (i : ι) => f i a)))
+Case conversion may be inaccurate. Consider using '#align finset.strongly_measurable_prod Finset.stronglyMeasurable_prodₓ'. -/
@[to_additive]
theorem Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i in s, f i a := by
@@ -608,6 +840,12 @@ theorem Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s :
end CommMonoid
+/- warning: measure_theory.strongly_measurable.is_separable_range -> MeasureTheory.StronglyMeasurable.isSeparable_range is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (TopologicalSpace.IsSeparable.{u2} β _inst_2 (Set.range.{u2, succ u1} β α f))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (TopologicalSpace.IsSeparable.{u1} β _inst_2 (Set.range.{u1, succ u2} β α f))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_rangeₓ'. -/
/-- The range of a strongly measurable function is separable. -/
theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : StronglyMeasurable f) :
TopologicalSpace.IsSeparable (range f) :=
@@ -622,6 +860,12 @@ theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : S
exact mem_range_self _
#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_range
+/- warning: measure_theory.strongly_measurable.separable_space_range_union_singleton -> MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {b : β}, TopologicalSpace.SeparableSpace.{u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) (Union.union.{u2} (Set.{u2} β) (Set.hasUnion.{u2} β) (Set.range.{u2, succ u1} β α f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b))) (Subtype.topologicalSpace.{u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x (Union.union.{u2} (Set.{u2} β) (Set.hasUnion.{u2} β) (Set.range.{u2, succ u1} β α f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b))) _inst_2))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (forall {b : β}, TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} β (Union.union.{u1} (Set.{u1} β) (Set.instUnionSet.{u1} β) (Set.range.{u1, succ u2} β α f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) b))) (instTopologicalSpaceSubtype.{u1} β (fun (x : β) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x (Union.union.{u1} (Set.{u1} β) (Set.instUnionSet.{u1} β) (Set.range.{u1, succ u2} β α f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) b))) _inst_2))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.separable_space_range_union_singleton MeasureTheory.StronglyMeasurable.separableSpace_range_union_singletonₓ'. -/
theorem separableSpace_range_union_singleton {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (hf : StronglyMeasurable f) {b : β} :
SeparableSpace (range f ∪ {b} : Set β) :=
@@ -635,6 +879,7 @@ variable {mα : MeasurableSpace α} [MeasurableSpace β]
include mα
+#print Measurable.stronglyMeasurable /-
/-- In a space with second countable topology, measurable implies strongly measurable. -/
theorem Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
[SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
@@ -648,20 +893,31 @@ theorem Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpa
⟨simple_func.approx_on f hf Set.univ default (Set.mem_univ _), fun x =>
simple_func.tendsto_approx_on hf (Set.mem_univ _) (by simp)⟩
#align measurable.strongly_measurable Measurable.stronglyMeasurable
+-/
+#print stronglyMeasurable_iff_measurable /-
/-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
theorem stronglyMeasurable_iff_measurable [TopologicalSpace β] [MetrizableSpace β] [BorelSpace β]
[SecondCountableTopology β] : StronglyMeasurable f ↔ Measurable f :=
⟨fun h => h.Measurable, fun h => Measurable.stronglyMeasurable h⟩
#align strongly_measurable_iff_measurable stronglyMeasurable_iff_measurable
+-/
+#print stronglyMeasurable_id /-
theorem stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
[OpensMeasurableSpace α] [SecondCountableTopology α] : StronglyMeasurable (id : α → α) :=
measurable_id.StronglyMeasurable
#align strongly_measurable_id stronglyMeasurable_id
+-/
end SecondCountableStronglyMeasurable
+/- warning: strongly_measurable_iff_measurable_separable -> stronglyMeasurable_iff_measurable_separable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : BorelSpace.{u2} β _inst_2 _inst_4], Iff (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (And (Measurable.{u1, u2} α β m _inst_4 f) (TopologicalSpace.IsSeparable.{u2} β _inst_2 (Set.range.{u2, succ u1} β α f)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : BorelSpace.{u1} β _inst_2 _inst_4], Iff (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (And (Measurable.{u2, u1} α β m _inst_4 f) (TopologicalSpace.IsSeparable.{u1} β _inst_2 (Set.range.{u1, succ u2} β α f)))
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separableₓ'. -/
/-- A function is strongly measurable if and only if it is measurable and has separable
range. -/
theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [TopologicalSpace β]
@@ -693,6 +949,12 @@ theorem stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α} [To
exact continuous_subtype_coe.comp_strongly_measurable g_smeas
#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separable
+/- warning: continuous.strongly_measurable -> Continuous.stronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : OpensMeasurableSpace.{u1} α _inst_3 _inst_2] {β : Type.{u2}} [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_5] [h : SecondCountableTopologyEither.{u1, u2} α β _inst_3 _inst_5] {f : α -> β}, (Continuous.{u1, u2} α β _inst_3 _inst_5 f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_5 _inst_2 f)
+but is expected to have type
+ forall {α : Type.{u2}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : OpensMeasurableSpace.{u2} α _inst_3 _inst_2] {β : Type.{u1}} [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_5] [h : SecondCountableTopologyEither.{u2, u1} α β _inst_3 _inst_5] {f : α -> β}, (Continuous.{u2, u1} α β _inst_3 _inst_5 f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_5 _inst_2 f)
+Case conversion may be inaccurate. Consider using '#align continuous.strongly_measurable Continuous.stronglyMeasurableₓ'. -/
/-- A continuous function is strongly measurable when either the source space or the target space
is second-countable. -/
theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
@@ -708,6 +970,12 @@ theorem Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
· exact hf.measurable.strongly_measurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
+/- warning: embedding.comp_strongly_measurable_iff -> Embedding.comp_stronglyMeasurable_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : TopologicalSpace.{u3} γ] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u3} γ _inst_4] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u3} β γ _inst_2 _inst_4 g) -> (Iff (MeasureTheory.StronglyMeasurable.{u1, u3} α γ _inst_4 m (fun (x : α) => g (f x))) (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_4 : TopologicalSpace.{u1} γ] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} γ _inst_4] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u1} β γ _inst_2 _inst_4 g) -> (Iff (MeasureTheory.StronglyMeasurable.{u3, u1} α γ _inst_4 m (fun (x : α) => g (f x))) (MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_2 m f))
+Case conversion may be inaccurate. Consider using '#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iffₓ'. -/
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
@@ -734,6 +1002,12 @@ theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [Topologi
simp [hg.inj.eq_iff]
#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff
+/- warning: strongly_measurable_of_tendsto -> stronglyMeasurable_of_tendsto is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] (u : Filter.{u3} ι) [_inst_4 : Filter.NeBot.{u3} ι u] [_inst_5 : Filter.IsCountablyGenerated.{u3} ι u] {f : ι -> α -> β} {g : α -> β}, (forall (i : ι), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (f i)) -> (Filter.Tendsto.{u3, max u1 u2} ι (α -> β) f u (nhds.{max u1 u2} (α -> β) (Pi.topologicalSpace.{u1, u2} α (fun (ᾰ : α) => β) (fun (a : α) => _inst_2)) g)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {ι : Type.{u3}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] (u : Filter.{u3} ι) [_inst_4 : Filter.NeBot.{u3} ι u] [_inst_5 : Filter.IsCountablyGenerated.{u3} ι u] {f : ι -> α -> β} {g : α -> β}, (forall (i : ι), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (f i)) -> (Filter.Tendsto.{u3, max u2 u1} ι (α -> β) f u (nhds.{max u2 u1} (α -> β) (Pi.topologicalSpace.{u2, u1} α (fun (ᾰ : α) => β) (fun (a : α) => _inst_2)) g)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m g)
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendstoₓ'. -/
/-- A sequential limit of strongly measurable functions is strongly measurable. -/
theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β}
@@ -754,6 +1028,12 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
exact mem_range_self _
#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendsto
+/- warning: measure_theory.strongly_measurable.piecewise -> MeasureTheory.StronglyMeasurable.piecewise is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {g : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{u1} α} {_x : DecidablePred.{succ u1} α (fun (_x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) _x s)}, (MeasurableSet.{u1} α m s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (Set.piecewise.{u1, succ u2} α (fun (ᾰ : α) => β) s f g (fun (j : α) => _x j)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {g : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {s : Set.{u2} α} {_x : DecidablePred.{succ u2} α (fun (_x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) _x s)}, (MeasurableSet.{u2} α m s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m g) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (Set.piecewise.{u2, succ u1} α (fun (ᾰ : α) => β) s f g (fun (j : α) => _x j)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewiseₓ'. -/
protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
{_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) :=
@@ -764,6 +1044,12 @@ protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s :
· simpa [hx] using hg.tendsto_approx x
#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewise
+/- warning: measure_theory.strongly_measurable.ite -> MeasureTheory.StronglyMeasurable.ite is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {g : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {p : α -> Prop} {_x : DecidablePred.{succ u1} α p}, (MeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => p a))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => ite.{succ u2} β (p x) (_x x) (f x) (g x)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {g : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {p : α -> Prop} {_x : DecidablePred.{succ u2} α p}, (MeasurableSet.{u2} α m (setOf.{u2} α (fun (a : α) => p a))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m g) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (fun (x : α) => ite.{succ u1} β (p x) (_x x) (f x) (g x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.iteₓ'. -/
/-- this is slightly different from `strongly_measurable.piecewise`. It can be used to show
`strongly_measurable (ite (x=0) 0 1)` by
`exact strongly_measurable.ite (measurable_set_singleton 0) strongly_measurable_const
@@ -775,6 +1061,12 @@ protected theorem ite {m : MeasurableSpace α} [TopologicalSpace β] {p : α →
StronglyMeasurable.piecewise hp hf hg
#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite
+/- warning: strongly_measurable_of_strongly_measurable_union_cover -> stronglyMeasurable_of_stronglyMeasurable_union_cover is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} (s : Set.{u1} α) (t : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (MeasurableSet.{u1} α m t) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Set.univ.{u1} α) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) β _inst_2 (Subtype.instMeasurableSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) m) (fun (a : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) => f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) a))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) β _inst_2 (Subtype.instMeasurableSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) m) (fun (a : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) => f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) t) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t))))) a))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} (s : Set.{u2} α) (t : Set.{u2} α), (MeasurableSet.{u2} α m s) -> (MeasurableSet.{u2} α m t) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (Set.univ.{u2} α) (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} (Set.Elem.{u2} α s) β _inst_2 (Subtype.instMeasurableSpace.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) m) (fun (a : Set.Elem.{u2} α s) => f (Subtype.val.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) a))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} (Set.Elem.{u2} α t) β _inst_2 (Subtype.instMeasurableSpace.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x t) m) (fun (a : Set.Elem.{u2} α t) => f (Subtype.val.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x t) a))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f)
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_coverₓ'. -/
theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
@@ -821,6 +1113,12 @@ theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpac
simp only [dif_neg hy, simple_func.apply_mk]
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
+/- warning: strongly_measurable_of_restrict_of_restrict_compl -> stronglyMeasurable_of_restrict_of_restrict_compl is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) β _inst_2 (Subtype.instMeasurableSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) m) (Set.restrict.{u1, u2} α (fun (ᾰ : α) => β) s f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s)) β _inst_2 (Subtype.instMeasurableSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s)) m) (Set.restrict.{u1, u2} α (fun (ᾰ : α) => β) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s) f)) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α}, (MeasurableSet.{u2} α m s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} (Set.Elem.{u2} α s) β _inst_2 (Subtype.instMeasurableSpace.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) m) (Set.restrict.{u2, u1} α (fun (ᾰ : α) => β) s f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} (Set.Elem.{u2} α (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) s)) β _inst_2 (Subtype.instMeasurableSpace.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) s)) m) (Set.restrict.{u2, u1} α (fun (ᾰ : α) => β) (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) s) f)) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f)
+Case conversion may be inaccurate. Consider using '#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_complₓ'. -/
theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
(h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
@@ -829,38 +1127,80 @@ theorem stronglyMeasurable_of_restrict_of_restrict_compl {m : MeasurableSpace α
h₂
#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl
+/- warning: measure_theory.strongly_measurable.indicator -> MeasureTheory.StronglyMeasurable.indicator is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β], (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (Set.indicator.{u1, u2} α β _inst_3 s f)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Zero.{u1} β], (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (forall {s : Set.{u2} α}, (MeasurableSet.{u2} α m s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (Set.indicator.{u2, u1} α β _inst_3 s f)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicatorₓ'. -/
protected theorem indicator {m : MeasurableSpace α} [TopologicalSpace β] [Zero β]
(hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
StronglyMeasurable (s.indicator f) :=
hf.piecewise hs stronglyMeasurable_const
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
+/- warning: measure_theory.strongly_measurable.dist -> MeasureTheory.StronglyMeasurable.dist is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) m g) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m (fun (x : α) => Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (g x)))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {β : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} β] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β _inst_2)) m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β _inst_2)) m g) -> (MeasureTheory.StronglyMeasurable.{u2, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m (fun (x : α) => Dist.dist.{u1} β (PseudoMetricSpace.toDist.{u1} β _inst_2) (f x) (g x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.distₓ'. -/
protected theorem dist {m : MeasurableSpace α} {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist
+/- warning: measure_theory.strongly_measurable.norm -> MeasureTheory.StronglyMeasurable.norm is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {β : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m (fun (x : α) => Norm.norm.{u2} β (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_2) (f x)))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {β : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} β _inst_2))) m f) -> (MeasureTheory.StronglyMeasurable.{u2, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m (fun (x : α) => Norm.norm.{u1} β (SeminormedAddCommGroup.toNorm.{u1} β _inst_2) (f x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.normₓ'. -/
protected theorem norm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ :=
continuous_norm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm
+/- warning: measure_theory.strongly_measurable.nnnorm -> MeasureTheory.StronglyMeasurable.nnnorm is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {β : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_2)) (f x)))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {β : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} β _inst_2))) m f) -> (MeasureTheory.StronglyMeasurable.{u2, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => NNNorm.nnnorm.{u1} β (SeminormedAddGroup.toNNNorm.{u1} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} β _inst_2)) (f x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnormₓ'. -/
protected theorem nnnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ :=
continuous_nnnorm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm
+/- warning: measure_theory.strongly_measurable.ennnorm -> MeasureTheory.StronglyMeasurable.ennnorm is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {β : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (fun (a : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_2)) (f a))))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {β : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} β] {f : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} β _inst_2))) m f) -> (Measurable.{u2, 0} α ENNReal m ENNReal.measurableSpace (fun (a : α) => ENNReal.some (NNNorm.nnnorm.{u1} β (SeminormedAddGroup.toNNNorm.{u1} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} β _inst_2)) (f a))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnormₓ'. -/
protected theorem ennnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
(ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).Measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
+/- warning: measure_theory.strongly_measurable.real_to_nnreal -> MeasureTheory.StronglyMeasurable.real_toNNReal is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {f : α -> Real}, (MeasureTheory.StronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => Real.toNNReal (f x)))
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {f : α -> Real}, (MeasureTheory.StronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f) -> (MeasureTheory.StronglyMeasurable.{u1, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => Real.toNNReal (f x)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNRealₓ'. -/
protected theorem real_toNNReal {m : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => (f x).toNNReal :=
continuous_real_toNNReal.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal
+/- warning: measurable_embedding.strongly_measurable_extend -> MeasurableEmbedding.stronglyMeasurable_extend is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β} {g : α -> γ} {g' : γ -> β} {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} [_inst_2 : TopologicalSpace.{u2} β], (MeasurableEmbedding.{u1, u3} α γ mα mγ g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 mγ g') -> (MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.extend.{succ u1, succ u3, succ u2} α γ β g f g'))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u1}} {γ : Type.{u2}} {f : α -> β} {g : α -> γ} {g' : γ -> β} {mα : MeasurableSpace.{u3} α} {mγ : MeasurableSpace.{u2} γ} [_inst_2 : TopologicalSpace.{u1} β], (MeasurableEmbedding.{u3, u2} α γ mα mγ g) -> (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 mα f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} γ β _inst_2 mγ g') -> (MeasureTheory.StronglyMeasurable.{u2, u1} γ β _inst_2 mγ (Function.extend.{succ u3, succ u2, succ u1} α γ β g f g'))
+Case conversion may be inaccurate. Consider using '#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extendₓ'. -/
theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
@@ -877,6 +1217,12 @@ theorem MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α
hg'.tendsto_approx x
#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
+/- warning: measurable_embedding.exists_strongly_measurable_extend -> MeasurableEmbedding.exists_stronglyMeasurable_extend is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β} {g : α -> γ} {mα : MeasurableSpace.{u1} α} {mγ : MeasurableSpace.{u3} γ} [_inst_2 : TopologicalSpace.{u2} β], (MeasurableEmbedding.{u1, u3} α γ mα mγ g) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 mα f) -> (γ -> (Nonempty.{succ u2} β)) -> (Exists.{max (succ u3) (succ u2)} (γ -> β) (fun (f' : γ -> β) => And (MeasureTheory.StronglyMeasurable.{u3, u2} γ β _inst_2 mγ f') (Eq.{max (succ u1) (succ u2)} (α -> β) (Function.comp.{succ u1, succ u3, succ u2} α γ β f' g) f)))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u1}} {γ : Type.{u2}} {f : α -> β} {g : α -> γ} {mα : MeasurableSpace.{u3} α} {mγ : MeasurableSpace.{u2} γ} [_inst_2 : TopologicalSpace.{u1} β], (MeasurableEmbedding.{u3, u2} α γ mα mγ g) -> (MeasureTheory.StronglyMeasurable.{u3, u1} α β _inst_2 mα f) -> (γ -> (Nonempty.{succ u1} β)) -> (Exists.{max (succ u1) (succ u2)} (γ -> β) (fun (f' : γ -> β) => And (MeasureTheory.StronglyMeasurable.{u2, u1} γ β _inst_2 mγ f') (Eq.{max (succ u3) (succ u1)} (α -> β) (Function.comp.{succ u3, succ u2, succ u1} α γ β f' g) f)))
+Case conversion may be inaccurate. Consider using '#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extendₓ'. -/
theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
@@ -886,6 +1232,12 @@ theorem MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g
funext fun x => hg.Injective.extend_apply _ _ _⟩
#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend
+/- warning: measure_theory.strongly_measurable.measurable_set_eq_fun -> MeasureTheory.StronglyMeasurable.measurableSet_eq_fun is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {E : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} E] [_inst_3 : TopologicalSpace.MetrizableSpace.{u2} E _inst_2] {f : α -> E} {g : α -> E}, (MeasureTheory.StronglyMeasurable.{u1, u2} α E _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α E _inst_2 m g) -> (MeasurableSet.{u1} α m (setOf.{u1} α (fun (x : α) => Eq.{succ u2} E (f x) (g x))))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {E : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} E] [_inst_3 : TopologicalSpace.MetrizableSpace.{u1} E _inst_2] {f : α -> E} {g : α -> E}, (MeasureTheory.StronglyMeasurable.{u2, u1} α E _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α E _inst_2 m g) -> (MeasurableSet.{u2} α m (setOf.{u2} α (fun (x : α) => Eq.{succ u1} E (f x) (g x))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_funₓ'. -/
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
{f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
MeasurableSet { x | f x = g x } := by
@@ -893,6 +1245,12 @@ theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [
exact (hf.prod_mk hg).Measurable is_closed_diagonal.measurable_set
#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
+/- warning: measure_theory.strongly_measurable.measurable_set_lt -> MeasureTheory.StronglyMeasurable.measurableSet_lt is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : LinearOrder.{u2} β] [_inst_4 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_3))))] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g) -> (MeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_3))))) (f a) (g a))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : LinearOrder.{u1} β] [_inst_4 : OrderClosedTopology.{u1} β _inst_2 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (DistribLattice.toLattice.{u1} β (instDistribLattice.{u1} β _inst_3)))))] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m g) -> (MeasurableSet.{u2} α m (setOf.{u2} α (fun (a : α) => LT.lt.{u1} β (Preorder.toLT.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (DistribLattice.toLattice.{u1} β (instDistribLattice.{u1} β _inst_3)))))) (f a) (g a))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_ltₓ'. -/
theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } :=
@@ -901,6 +1259,12 @@ theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearO
exact (hf.prod_mk hg).Measurable is_open_lt_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_lt
+/- warning: measure_theory.strongly_measurable.measurable_set_le -> MeasureTheory.StronglyMeasurable.measurableSet_le is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Preorder.{u2} β] [_inst_4 : OrderClosedTopology.{u2} β _inst_2 _inst_3] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m g) -> (MeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LE.le.{u2} β (Preorder.toHasLe.{u2} β _inst_3) (f a) (g a))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Preorder.{u1} β] [_inst_4 : OrderClosedTopology.{u1} β _inst_2 _inst_3] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m g) -> (MeasurableSet.{u2} α m (setOf.{u2} α (fun (a : α) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_3) (f a) (g a))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_leₓ'. -/
theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
[OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : MeasurableSet { a | f a ≤ g a } :=
@@ -909,6 +1273,12 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
+/- warning: measure_theory.strongly_measurable.strongly_measurable_in_set -> MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] {s : Set.{u1} α} {f : α -> β}, (MeasurableSet.{u1} α m s) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) -> (forall (x : α), (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_3))))) -> (Exists.{max 1 (succ u1) (succ u2)} (Nat -> (MeasureTheory.SimpleFunc.{u1, u2} α m β)) (fun (fs : Nat -> (MeasureTheory.SimpleFunc.{u1, u2} α m β)) => And (forall (x : α), Filter.Tendsto.{0, u2} Nat β (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (fs n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u2} β _inst_2 (f x))) (forall (x : α), (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)) -> (forall (n : Nat), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) (fs n) x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_3)))))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Zero.{u1} β] {s : Set.{u2} α} {f : α -> β}, (MeasurableSet.{u2} α m s) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) -> (forall (x : α), (Not (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s)) -> (Eq.{succ u1} β (f x) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_3)))) -> (Exists.{max (succ u2) (succ u1)} (Nat -> (MeasureTheory.SimpleFunc.{u2, u1} α m β)) (fun (fs : Nat -> (MeasureTheory.SimpleFunc.{u2, u1} α m β)) => And (forall (x : α), Filter.Tendsto.{0, u1} Nat β (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β (fs n) x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} β _inst_2 (f x))) (forall (x : α), (Not (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s)) -> (forall (n : Nat), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β (fs n) x) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_3))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_setₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
@@ -935,6 +1305,12 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
exact tendsto_const_nhds
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
+/- warning: measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on -> MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {E : Type.{u2}} {m : MeasurableSpace.{u1} α} {m₂ : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} E] [_inst_3 : Zero.{u2} E] {s : Set.{u1} α} {f : α -> E}, (MeasurableSet.{u1} α m s) -> (forall (t : Set.{u1} α), (MeasurableSet.{u1} α m (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) -> (MeasurableSet.{u1} α m₂ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α E _inst_2 m f) -> (forall (x : α), (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)) -> (Eq.{succ u2} E (f x) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E _inst_3))))) -> (MeasureTheory.StronglyMeasurable.{u1, u2} α E _inst_2 m₂ f)
+but is expected to have type
+ forall {α : Type.{u2}} {E : Type.{u1}} {m : MeasurableSpace.{u2} α} {m₂ : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} E] [_inst_3 : Zero.{u1} E] {s : Set.{u2} α} {f : α -> E}, (MeasurableSet.{u2} α m s) -> (forall (t : Set.{u2} α), (MeasurableSet.{u2} α m (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) s t)) -> (MeasurableSet.{u2} α m₂ (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) s t))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α E _inst_2 m f) -> (forall (x : α), (Not (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s)) -> (Eq.{succ u1} E (f x) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E _inst_3)))) -> (MeasureTheory.StronglyMeasurable.{u2, u1} α E _inst_2 m₂ f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_onₓ'. -/
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
@@ -981,6 +1357,12 @@ theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableS
exact hg_seq_tendsto x
#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
+/- warning: measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le -> MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : SeminormedAddCommGroup.{u2} β] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m m0), (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (forall (μ : MeasureTheory.Measure.{u1} α m0) [_inst_3 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)], Exists.{succ u1} (Nat -> (Set.{u1} α)) (fun (s : Nat -> (Set.{u1} α)) => And (forall (n : Nat), And (MeasurableSet.{u1} α m (s n)) (And (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m0) (fun (_x : MeasureTheory.Measure.{u1} α m0) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m0) μ (s n)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (s n)) -> (LE.le.{0} Real Real.hasLe (Norm.norm.{u2} β (SeminormedAddCommGroup.toHasNorm.{u2} β _inst_2) (f x)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))))) (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (i : Nat) => s i)) (Set.univ.{u1} α))))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : SeminormedAddCommGroup.{u2} β] {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instLEMeasurableSpace.{u1} α) m m0), (MeasureTheory.StronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_2))) m f) -> (forall (μ : MeasureTheory.Measure.{u1} α m0) [_inst_3 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)], Exists.{succ u1} (Nat -> (Set.{u1} α)) (fun (s : Nat -> (Set.{u1} α)) => And (forall (n : Nat), And (MeasurableSet.{u1} α m (s n)) (And (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m0 μ) (s n)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (s n)) -> (LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} β (SeminormedAddCommGroup.toNorm.{u2} β _inst_2) (f x)) (Nat.cast.{0} Real Real.natCast n))))) (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (i : Nat) => s i)) (Set.univ.{u1} α))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_leₓ'. -/
/-- If a function `f` is strongly measurable w.r.t. a sub-σ-algebra `m` and the measure is σ-finite
on `m`, then there exists spanning measurable sets with finite measure on which `f` has bounded
norm. In particular, `f` is integrable on each of those sets. -/
@@ -1023,6 +1405,12 @@ end StronglyMeasurable
/-! ## Finitely strongly measurable functions -/
+/- warning: measure_theory.fin_strongly_measurable_zero -> MeasureTheory.finStronglyMeasurable_zero is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β], MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_2))))) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : Zero.{u1} β] [_inst_3 : TopologicalSpace.{u1} β], MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.11808 : α) => β) (fun (i : α) => _inst_2)))) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zeroₓ'. -/
theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
[TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ :=
⟨0, by
@@ -1035,37 +1423,57 @@ namespace FinStronglyMeasurable
variable {m0 : MeasurableSpace α} {μ : Measure α} {f g : α → β}
-theorem aeFinStronglyMeasurable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ) :
- AeFinStronglyMeasurable f μ :=
+#print MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable /-
+theorem aefinStronglyMeasurable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ) :
+ AEFinStronglyMeasurable f μ :=
⟨f, hf, ae_eq_refl f⟩
-#align measure_theory.fin_strongly_measurable.ae_fin_strongly_measurable MeasureTheory.FinStronglyMeasurable.aeFinStronglyMeasurable
+#align measure_theory.fin_strongly_measurable.ae_fin_strongly_measurable MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable
+-/
section sequence
variable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ)
+#print MeasureTheory.FinStronglyMeasurable.approx /-
/-- A sequence of simple functions such that `∀ x, tendsto (λ n, hf.approx n x) at_top (𝓝 (f x))`
and `∀ n, μ (support (hf.approx n)) < ∞`. These properties are given by
`fin_strongly_measurable.tendsto_approx` and `fin_strongly_measurable.fin_support_approx`. -/
protected noncomputable def approx : ℕ → α →ₛ β :=
hf.some
#align measure_theory.fin_strongly_measurable.approx MeasureTheory.FinStronglyMeasurable.approx
+-/
+/- warning: measure_theory.fin_strongly_measurable.fin_support_approx -> MeasureTheory.FinStronglyMeasurable.fin_support_approx is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β] (hf : MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m0 f μ) (n : Nat), LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m0) (fun (_x : MeasureTheory.Measure.{u1} α m0) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m0) μ (Function.support.{u1, u2} α β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m0 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m0 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m0) (MeasureTheory.FinStronglyMeasurable.approx.{u1, u2} α β m0 μ f _inst_2 _inst_3 hf n)))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m0 : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m0} {f : α -> β} [_inst_2 : Zero.{u1} β] [_inst_3 : TopologicalSpace.{u1} β] (hf : MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 m0 f μ) (n : Nat), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m0 μ) (Function.support.{u2, u1} α β _inst_2 (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m0 β (MeasureTheory.FinStronglyMeasurable.approx.{u2, u1} α β m0 μ f _inst_2 _inst_3 hf n)))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approxₓ'. -/
protected theorem fin_support_approx : ∀ n, μ (support (hf.approx n)) < ∞ :=
hf.choose_spec.1
#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approx
+#print MeasureTheory.FinStronglyMeasurable.tendsto_approx /-
protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
hf.choose_spec.2
#align measure_theory.fin_strongly_measurable.tendsto_approx MeasureTheory.FinStronglyMeasurable.tendsto_approx
+-/
end sequence
+#print MeasureTheory.FinStronglyMeasurable.stronglyMeasurable /-
protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
(hf : FinStronglyMeasurable f μ) : StronglyMeasurable f :=
⟨hf.approx, hf.tendsto_approx⟩
#align measure_theory.fin_strongly_measurable.strongly_measurable MeasureTheory.FinStronglyMeasurable.stronglyMeasurable
+-/
+/- warning: measure_theory.fin_strongly_measurable.exists_set_sigma_finite -> MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m0 f μ) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m0 t) (And (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2))))) (MeasureTheory.SigmaFinite.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ t)))))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m0 f μ) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m0 t) (And (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_2)))) (MeasureTheory.SigmaFinite.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ t)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFiniteₓ'. -/
theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
(hf : FinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) :=
@@ -1091,16 +1499,24 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
exact Set.compl_union_self _
#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
+#print MeasureTheory.FinStronglyMeasurable.measurable /-
/-- A finitely strongly measurable function is measurable. -/
protected theorem measurable [Zero β] [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : FinStronglyMeasurable f μ) : Measurable f :=
hf.StronglyMeasurable.Measurable
#align measure_theory.fin_strongly_measurable.measurable MeasureTheory.FinStronglyMeasurable.measurable
+-/
section Arithmetic
variable [TopologicalSpace β]
+/- warning: measure_theory.fin_strongly_measurable.mul -> MeasureTheory.FinStronglyMeasurable.mul is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m0 (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m0 (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mulₓ'. -/
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ :=
by
@@ -1111,6 +1527,12 @@ protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMe
exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul
+/- warning: measure_theory.fin_strongly_measurable.add -> MeasureTheory.FinStronglyMeasurable.add is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m0 (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m0 (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.addₓ'. -/
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ :=
⟨fun n => hf.approx n + hg.approx n, fun n =>
@@ -1120,6 +1542,12 @@ protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasura
fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add
+/- warning: measure_theory.fin_strongly_measurable.neg -> MeasureTheory.FinStronglyMeasurable.neg is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasNeg.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))) f) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => NegZeroClass.toNeg.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3))))) f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.negₓ'. -/
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
FinStronglyMeasurable (-f) μ :=
by
@@ -1129,6 +1557,12 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMe
exact hf.fin_support_approx n
#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg
+/- warning: measure_theory.fin_strongly_measurable.sub -> MeasureTheory.FinStronglyMeasurable.sub is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m0 (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m0 (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.subₓ'. -/
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ :=
⟨fun n => hf.approx n - hg.approx n, fun n =>
@@ -1138,6 +1572,12 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
+/- warning: measure_theory.fin_strongly_measurable.const_smul -> MeasureTheory.FinStronglyMeasurable.const_smul is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m0 f μ) -> (forall (c : 𝕜), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m0 (SMul.smul.{u3, max u1 u2} 𝕜 (α -> β) (Function.hasSMul.{u1, u3, u2} α 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6)))) c f) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m0 f μ) -> (forall (c : 𝕜), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m0 (HSMul.hSMul.{u3, max u1 u2, max u1 u2} 𝕜 (α -> β) (α -> β) (instHSMul.{u3, max u1 u2} 𝕜 (α -> β) (Pi.instSMul.{u1, u2, u3} α 𝕜 (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13158 : α) => β) (fun (i : α) => SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))))) c f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smulₓ'. -/
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
FinStronglyMeasurable (c • f) μ :=
@@ -1153,6 +1593,12 @@ section Order
variable [TopologicalSpace β] [Zero β]
+/- warning: measure_theory.fin_strongly_measurable.sup -> MeasureTheory.FinStronglyMeasurable.sup is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.supₓ'. -/
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ :=
by
@@ -1163,6 +1609,12 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
+/- warning: measure_theory.fin_strongly_measurable.inf -> MeasureTheory.FinStronglyMeasurable.inf is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} {f : α -> β} {g : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 f μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 g μ) -> (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m0 (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.infₓ'. -/
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ :=
by
@@ -1177,6 +1629,12 @@ end Order
end FinStronglyMeasurable
+/- warning: measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite -> MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {f : α -> β} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T2Space.{u2} β _inst_2] [_inst_4 : Zero.{u2} β] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Iff (MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_4 m f μ) (And (MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m f) (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m t) (And (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)) -> (Eq.{succ u2} β (f x) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_4))))) (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t))))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {f : α -> β} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : T2Space.{u1} β _inst_2] [_inst_4 : Zero.{u1} β] {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m}, Iff (MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_4 m f μ) (And (MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m f) (Exists.{succ u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (MeasurableSet.{u2} α m t) (And (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) t)) -> (Eq.{succ u1} β (f x) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_4)))) (MeasureTheory.SigmaFinite.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ t))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFiniteₓ'. -/
theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β}
[TopologicalSpace β] [T2Space β] [Zero β] {m : MeasurableSpace α} {μ : Measure α} :
FinStronglyMeasurable f μ ↔
@@ -1187,51 +1645,93 @@ theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
hf.2.choose_spec.2.2⟩
#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
-theorem aeFinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
- [TopologicalSpace β] : AeFinStronglyMeasurable (0 : α → β) μ :=
+/- warning: measure_theory.ae_fin_strongly_measurable_zero -> MeasureTheory.aefinStronglyMeasurable_zero is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) [_inst_2 : Zero.{u2} β] [_inst_3 : TopologicalSpace.{u2} β], MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_2))))) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m) [_inst_2 : Zero.{u1} β] [_inst_3 : TopologicalSpace.{u1} β], MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13622 : α) => β) (fun (i : α) => _inst_2)))) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aefinStronglyMeasurable_zeroₓ'. -/
+theorem aefinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
+ [TopologicalSpace β] : AEFinStronglyMeasurable (0 : α → β) μ :=
⟨0, finStronglyMeasurable_zero, EventuallyEq.rfl⟩
-#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aeFinStronglyMeasurable_zero
+#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aefinStronglyMeasurable_zero
/-! ## Almost everywhere strongly measurable functions -/
-theorem aeStronglyMeasurable_const {α β} {m : MeasurableSpace α} {μ : Measure α}
- [TopologicalSpace β] {b : β} : AeStronglyMeasurable (fun a : α => b) μ :=
- stronglyMeasurable_const.AeStronglyMeasurable
-#align measure_theory.ae_strongly_measurable_const MeasureTheory.aeStronglyMeasurable_const
-
+/- warning: measure_theory.ae_strongly_measurable_const -> MeasureTheory.aestronglyMeasurable_const is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {b : β}, MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (a : α) => b) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {b : β}, MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (fun (a : α) => b) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_constₓ'. -/
+theorem aestronglyMeasurable_const {α β} {m : MeasurableSpace α} {μ : Measure α}
+ [TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun a : α => b) μ :=
+ stronglyMeasurable_const.AEStronglyMeasurable
+#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_const
+
+/- warning: measure_theory.ae_strongly_measurable_one -> MeasureTheory.aestronglyMeasurable_one is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : One.{u2} β], MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (OfNat.ofNat.{max u1 u2} (α -> β) 1 (OfNat.mk.{max u1 u2} (α -> β) 1 (One.one.{max u1 u2} (α -> β) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3))))) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : One.{u1} β], MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (OfNat.ofNat.{max u2 u1} (α -> β) 1 (One.toOfNat1.{max u2 u1} (α -> β) (Pi.instOne.{u2, u1} α (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.13692 : α) => β) (fun (i : α) => _inst_3)))) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_oneₓ'. -/
@[to_additive]
-theorem aeStronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
- [One β] : AeStronglyMeasurable (1 : α → β) μ :=
- stronglyMeasurable_one.AeStronglyMeasurable
-#align measure_theory.ae_strongly_measurable_one MeasureTheory.aeStronglyMeasurable_one
-#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aeStronglyMeasurable_zero
-
+theorem aestronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
+ [One β] : AEStronglyMeasurable (1 : α → β) μ :=
+ stronglyMeasurable_one.AEStronglyMeasurable
+#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_one
+#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aestronglyMeasurable_zero
+
+/- warning: measure_theory.subsingleton.ae_strongly_measurable -> MeasureTheory.Subsingleton.aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Subsingleton.{succ u2} β] {μ : MeasureTheory.Measure.{u1} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Subsingleton.{succ u1} β] {μ : MeasureTheory.Measure.{u2} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurableₓ'. -/
@[simp]
-theorem Subsingleton.aeStronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
- [Subsingleton β] {μ : Measure α} (f : α → β) : AeStronglyMeasurable f μ :=
- (Subsingleton.stronglyMeasurable f).AeStronglyMeasurable
-#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aeStronglyMeasurable
-
+theorem Subsingleton.aestronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
+ [Subsingleton β] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
+ (Subsingleton.stronglyMeasurable f).AEStronglyMeasurable
+#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurable
+
+/- warning: measure_theory.subsingleton.ae_strongly_measurable' -> MeasureTheory.Subsingleton.aestronglyMeasurable' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : Subsingleton.{succ u1} α] {μ : MeasureTheory.Measure.{u1} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : Subsingleton.{succ u2} α] {μ : MeasureTheory.Measure.{u2} α m} (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'ₓ'. -/
@[simp]
-theorem Subsingleton.ae_strongly_measurable' {m : MeasurableSpace α} [TopologicalSpace β]
- [Subsingleton α] {μ : Measure α} (f : α → β) : AeStronglyMeasurable f μ :=
- (Subsingleton.strongly_measurable' f).AeStronglyMeasurable
-#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.ae_strongly_measurable'
-
+theorem Subsingleton.aestronglyMeasurable' {m : MeasurableSpace α} [TopologicalSpace β]
+ [Subsingleton α] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ :=
+ (Subsingleton.stronglyMeasurable' f).AEStronglyMeasurable
+#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'
+
+/- warning: measure_theory.ae_strongly_measurable_zero_measure -> MeasureTheory.aestronglyMeasurable_zero_measure is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_3 _inst_2 f (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_2) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α _inst_2) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α _inst_2) (MeasureTheory.Measure.instZero.{u1} α _inst_2))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : MeasurableSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] (f : α -> β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_3 _inst_2 f (OfNat.ofNat.{u2} (MeasureTheory.Measure.{u2} α _inst_2) 0 (Zero.toOfNat0.{u2} (MeasureTheory.Measure.{u2} α _inst_2) (MeasureTheory.Measure.instZero.{u2} α _inst_2)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measureₓ'. -/
@[simp]
-theorem aeStronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
- AeStronglyMeasurable f (0 : Measure α) :=
+theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
+ AEStronglyMeasurable f (0 : Measure α) :=
by
nontriviality α
inhabit α
exact ⟨fun x => f default, strongly_measurable_const, rfl⟩
-#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aeStronglyMeasurable_zero_measure
-
-theorem SimpleFunc.aeStronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
- (f : α →ₛ β) : AeStronglyMeasurable f μ :=
- f.StronglyMeasurable.AeStronglyMeasurable
-#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aeStronglyMeasurable
+#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measure
+
+/- warning: measure_theory.simple_func.ae_strongly_measurable -> MeasureTheory.SimpleFunc.aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α m β), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α m β), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aestronglyMeasurableₓ'. -/
+theorem SimpleFunc.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
+ (f : α →ₛ β) : AEStronglyMeasurable f μ :=
+ f.StronglyMeasurable.AEStronglyMeasurable
+#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aestronglyMeasurable
namespace AeStronglyMeasurable
@@ -1240,174 +1740,296 @@ variable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] [Topol
section Mk
+#print MeasureTheory.AEStronglyMeasurable.mk /-
/-- A `strongly_measurable` function such that `f =ᵐ[μ] hf.mk f`. See lemmas
`strongly_measurable_mk` and `ae_eq_mk`. -/
-protected noncomputable def mk (f : α → β) (hf : AeStronglyMeasurable f μ) : α → β :=
+protected noncomputable def mk (f : α → β) (hf : AEStronglyMeasurable f μ) : α → β :=
hf.some
-#align measure_theory.ae_strongly_measurable.mk MeasureTheory.AeStronglyMeasurable.mk
+#align measure_theory.ae_strongly_measurable.mk MeasureTheory.AEStronglyMeasurable.mk
+-/
-theorem stronglyMeasurable_mk (hf : AeStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) :=
+/- warning: measure_theory.ae_strongly_measurable.strongly_measurable_mk -> MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_2 m (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 f hf)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ), MeasureTheory.StronglyMeasurable.{u2, u1} α β _inst_2 m (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 f hf)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mkₓ'. -/
+theorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) :=
hf.choose_spec.1
-#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AeStronglyMeasurable.stronglyMeasurable_mk
+#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk
+#print MeasureTheory.AEStronglyMeasurable.measurable_mk /-
theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
- (hf : AeStronglyMeasurable f μ) : Measurable (hf.mk f) :=
+ (hf : AEStronglyMeasurable f μ) : Measurable (hf.mk f) :=
hf.stronglyMeasurable_mk.Measurable
-#align measure_theory.ae_strongly_measurable.measurable_mk MeasureTheory.AeStronglyMeasurable.measurable_mk
+#align measure_theory.ae_strongly_measurable.measurable_mk MeasureTheory.AEStronglyMeasurable.measurable_mk
+-/
-theorem ae_eq_mk (hf : AeStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
+/- warning: measure_theory.ae_strongly_measurable.ae_eq_mk -> MeasureTheory.AEStronglyMeasurable.ae_eq_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 f hf)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} (hf : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 f hf)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mkₓ'. -/
+theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
-#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AeStronglyMeasurable.ae_eq_mk
+#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk
-protected theorem aEMeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
- [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AeStronglyMeasurable f μ) :
+#print MeasureTheory.AEStronglyMeasurable.aemeasurable /-
+protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
+ [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEStronglyMeasurable f μ) :
AEMeasurable f μ :=
⟨hf.mk f, hf.stronglyMeasurable_mk.Measurable, hf.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.ae_measurable MeasureTheory.AeStronglyMeasurable.aEMeasurable
+#align measure_theory.ae_strongly_measurable.ae_measurable MeasureTheory.AEStronglyMeasurable.aemeasurable
+-/
end Mk
-theorem congr (hf : AeStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AeStronglyMeasurable g μ :=
+/- warning: measure_theory.ae_strongly_measurable.congr -> MeasureTheory.AEStronglyMeasurable.congr is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f g) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m g μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congrₓ'. -/
+theorem congr (hf : AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AEStronglyMeasurable g μ :=
⟨hf.mk f, hf.stronglyMeasurable_mk, h.symm.trans hf.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AeStronglyMeasurable.congr
-
-theorem aeStronglyMeasurable_congr (h : f =ᵐ[μ] g) :
- AeStronglyMeasurable f μ ↔ AeStronglyMeasurable g μ :=
+#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congr
+
+/- warning: ae_strongly_measurable_congr -> aestronglyMeasurable_congr is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β}, (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {g : α -> β}, (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m g μ))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_congr aestronglyMeasurable_congrₓ'. -/
+theorem aestronglyMeasurable_congr (h : f =ᵐ[μ] g) :
+ AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g μ :=
⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩
-#align ae_strongly_measurable_congr aeStronglyMeasurable_congr
-
-theorem mono_measure {ν : Measure α} (hf : AeStronglyMeasurable f μ) (h : ν ≤ μ) :
- AeStronglyMeasurable f ν :=
+#align ae_strongly_measurable_congr aestronglyMeasurable_congr
+
+/- warning: measure_theory.ae_strongly_measurable.mono_measure -> MeasureTheory.AEStronglyMeasurable.mono_measure is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ν : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toHasLe.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f ν)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ν : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (LE.le.{u2} (MeasureTheory.Measure.{u2} α m) (Preorder.toLE.{u2} (MeasureTheory.Measure.{u2} α m) (PartialOrder.toPreorder.{u2} (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.instPartialOrder.{u2} α m))) ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f ν)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measureₓ'. -/
+theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν ≤ μ) :
+ AEStronglyMeasurable f ν :=
⟨hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AeStronglyMeasurable.mono_measure
-
-protected theorem mono' {ν : Measure α} (h : AeStronglyMeasurable f μ) (h' : ν ≪ μ) :
- AeStronglyMeasurable f ν :=
+#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure
+
+/- warning: measure_theory.ae_strongly_measurable.mono' -> MeasureTheory.AEStronglyMeasurable.mono' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ν : MeasureTheory.Measure.{u1} α m}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u1} α m ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f ν)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ν : MeasureTheory.Measure.{u2} α m}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u2} α m ν μ) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f ν)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'ₓ'. -/
+protected theorem mono' {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
+ AEStronglyMeasurable f ν :=
⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AeStronglyMeasurable.mono'
-
-theorem mono_set {s t} (h : s ⊆ t) (ht : AeStronglyMeasurable f (μ.restrict t)) :
- AeStronglyMeasurable f (μ.restrict s) :=
+#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'
+
+/- warning: measure_theory.ae_strongly_measurable.mono_set -> MeasureTheory.AEStronglyMeasurable.mono_set is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ t)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α}, (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s t) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ t)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_setₓ'. -/
+theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
+ AEStronglyMeasurable f (μ.restrict s) :=
ht.mono_measure (restrict_mono h le_rfl)
-#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AeStronglyMeasurable.mono_set
-
-protected theorem restrict (hfm : AeStronglyMeasurable f μ) {s} :
- AeStronglyMeasurable f (μ.restrict s) :=
+#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_set
+
+/- warning: measure_theory.ae_strongly_measurable.restrict -> MeasureTheory.AEStronglyMeasurable.restrict is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (forall {s : Set.{u1} α}, MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (forall {s : Set.{u2} α}, MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrictₓ'. -/
+protected theorem restrict (hfm : AEStronglyMeasurable f μ) {s} :
+ AEStronglyMeasurable f (μ.restrict s) :=
hfm.mono_measure Measure.restrict_le_self
-#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AeStronglyMeasurable.restrict
-
-theorem ae_mem_imp_eq_mk {s} (h : AeStronglyMeasurable f (μ.restrict s)) :
+#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrict
+
+/- warning: measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk -> MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α} (h : MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ s)), Filter.Eventually.{u1} α (fun (x : α) => (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Eq.{succ u2} β (f x) (MeasureTheory.AEStronglyMeasurable.mk.{u1, u2} α β m (MeasureTheory.Measure.restrict.{u1} α m μ s) _inst_2 f h x))) (MeasureTheory.Measure.ae.{u1} α m μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α} (h : MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ s)), Filter.Eventually.{u2} α (fun (x : α) => (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) -> (Eq.{succ u1} β (f x) (MeasureTheory.AEStronglyMeasurable.mk.{u2, u1} α β m (MeasureTheory.Measure.restrict.{u2} α m μ s) _inst_2 f h x))) (MeasureTheory.Measure.ae.{u2} α m μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mkₓ'. -/
+theorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (μ.restrict s)) :
∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x :=
ae_imp_of_ae_restrict h.ae_eq_mk
-#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AeStronglyMeasurable.ae_mem_imp_eq_mk
-
+#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk
+
+/- warning: continuous.comp_ae_strongly_measurable -> Continuous.comp_aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u3, u2} β γ _inst_2 _inst_3 g) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 m (fun (x : α) => g (f x)) μ)
+Case conversion may be inaccurate. Consider using '#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurableₓ'. -/
/-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable. -/
-theorem Continuous.comp_aeStronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
- (hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => g (f x)) μ :=
+theorem Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
+ (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => g (f x)) μ :=
⟨_, hg.comp_stronglyMeasurable hf.stronglyMeasurable_mk, EventuallyEq.fun_comp hf.ae_eq_mk g⟩
-#align continuous.comp_ae_strongly_measurable Continuous.comp_aeStronglyMeasurable
-
+#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurable
+
+/- warning: continuous.ae_strongly_measurable -> Continuous.aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.{u1} α] [_inst_5 : OpensMeasurableSpace.{u1} α _inst_4 m] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_7 : SecondCountableTopologyEither.{u1, u2} α β _inst_4 _inst_2], (Continuous.{u1, u2} α β _inst_4 _inst_2 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_4 : TopologicalSpace.{u2} α] [_inst_5 : OpensMeasurableSpace.{u2} α _inst_4 m] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_7 : SecondCountableTopologyEither.{u2, u1} α β _inst_4 _inst_2], (Continuous.{u2, u1} α β _inst_4 _inst_2 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ)
+Case conversion may be inaccurate. Consider using '#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurableₓ'. -/
/-- A continuous function from `α` to `β` is ae strongly measurable when one of the two spaces is
second countable. -/
-theorem Continuous.aeStronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
+theorem Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
[PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] (hf : Continuous f) :
- AeStronglyMeasurable f μ :=
- hf.StronglyMeasurable.AeStronglyMeasurable
-#align continuous.ae_strongly_measurable Continuous.aeStronglyMeasurable
-
-protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AeStronglyMeasurable f μ)
- (hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (fun x => (f x, g x)) μ :=
+ AEStronglyMeasurable f μ :=
+ hf.StronglyMeasurable.AEStronglyMeasurable
+#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurable
+
+/- warning: measure_theory.ae_strongly_measurable.prod_mk -> MeasureTheory.AEStronglyMeasurable.prod_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u3} β γ (f x) (g x)) μ)
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {f : α -> β} {g : α -> γ}, (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} α γ _inst_3 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, max u1 u2} α (Prod.{u2, u1} β γ) (instTopologicalSpaceProd.{u2, u1} β γ _inst_2 _inst_3) m (fun (x : α) => Prod.mk.{u2, u1} β γ (f x) (g x)) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mkₓ'. -/
+protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
+ (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (fun x => (f x, g x)) μ :=
⟨fun x => (hf.mk f x, hg.mk g x), hf.stronglyMeasurable_mk.prod_mk hg.stronglyMeasurable_mk,
hf.ae_eq_mk.prod_mk hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AeStronglyMeasurable.prod_mk
-
+#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk
+
+/- warning: measurable.ae_strongly_measurable -> Measurable.aestronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_6 : TopologicalSpace.SecondCountableTopology.{u2} β _inst_2] [_inst_7 : OpensMeasurableSpace.{u2} β _inst_2 _inst_4], (Measurable.{u1, u2} α β m _inst_4 f) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] [_inst_6 : TopologicalSpace.SecondCountableTopology.{u1} β _inst_2] [_inst_7 : OpensMeasurableSpace.{u1} β _inst_2 _inst_4], (Measurable.{u2, u1} α β m _inst_4 f) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ)
+Case conversion may be inaccurate. Consider using '#align measurable.ae_strongly_measurable Measurable.aestronglyMeasurableₓ'. -/
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
-theorem Measurable.aeStronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [MeasurableSpace β]
+theorem Measurable.aestronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [MeasurableSpace β]
[PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β]
- (hf : Measurable f) : AeStronglyMeasurable f μ :=
- hf.StronglyMeasurable.AeStronglyMeasurable
-#align measurable.ae_strongly_measurable Measurable.aeStronglyMeasurable
+ (hf : Measurable f) : AEStronglyMeasurable f μ :=
+ hf.StronglyMeasurable.AEStronglyMeasurable
+#align measurable.ae_strongly_measurable Measurable.aestronglyMeasurable
section Arithmetic
+#print MeasureTheory.AEStronglyMeasurable.mul /-
@[to_additive]
-protected theorem mul [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ)
- (hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (f * g) μ :=
+protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
+ (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.stronglyMeasurable_mk.mul hg.stronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mul MeasureTheory.AeStronglyMeasurable.mul
-#align measure_theory.ae_strongly_measurable.add MeasureTheory.AeStronglyMeasurable.add
+#align measure_theory.ae_strongly_measurable.mul MeasureTheory.AEStronglyMeasurable.mul
+#align measure_theory.ae_strongly_measurable.add MeasureTheory.AEStronglyMeasurable.add
+-/
+#print MeasureTheory.AEStronglyMeasurable.mul_const /-
@[to_additive]
-protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
- AeStronglyMeasurable (fun x => f x * c) μ :=
- hf.mul aeStronglyMeasurable_const
-#align measure_theory.ae_strongly_measurable.mul_const MeasureTheory.AeStronglyMeasurable.mul_const
-#align measure_theory.ae_strongly_measurable.add_const MeasureTheory.AeStronglyMeasurable.add_const
+protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
+ AEStronglyMeasurable (fun x => f x * c) μ :=
+ hf.mul aestronglyMeasurable_const
+#align measure_theory.ae_strongly_measurable.mul_const MeasureTheory.AEStronglyMeasurable.mul_const
+#align measure_theory.ae_strongly_measurable.add_const MeasureTheory.AEStronglyMeasurable.add_const
+-/
+#print MeasureTheory.AEStronglyMeasurable.const_mul /-
@[to_additive]
-protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
- AeStronglyMeasurable (fun x => c * f x) μ :=
- aeStronglyMeasurable_const.mul hf
-#align measure_theory.ae_strongly_measurable.const_mul MeasureTheory.AeStronglyMeasurable.const_mul
-#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AeStronglyMeasurable.const_add
+protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
+ AEStronglyMeasurable (fun x => c * f x) μ :=
+ aestronglyMeasurable_const.mul hf
+#align measure_theory.ae_strongly_measurable.const_mul MeasureTheory.AEStronglyMeasurable.const_mul
+#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AEStronglyMeasurable.const_add
+-/
+/- warning: measure_theory.ae_strongly_measurable.inv -> MeasureTheory.AEStronglyMeasurable.inv is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inv.inv.{max u1 u2} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasInv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4))) f) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inv.inv.{max u2 u1} (α -> β) (Pi.instInv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => InvOneClass.toInv.{u2} β (DivInvOneMonoid.toInvOneClass.{u2} β (DivisionMonoid.toDivInvOneMonoid.{u2} β (Group.toDivisionMonoid.{u2} β _inst_4))))) f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.invₓ'. -/
@[to_additive]
-protected theorem inv [Group β] [TopologicalGroup β] (hf : AeStronglyMeasurable f μ) :
- AeStronglyMeasurable f⁻¹ μ :=
+protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
+ AEStronglyMeasurable f⁻¹ μ :=
⟨(hf.mk f)⁻¹, hf.stronglyMeasurable_mk.inv, hf.ae_eq_mk.inv⟩
-#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AeStronglyMeasurable.inv
-#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AeStronglyMeasurable.neg
-
+#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
+#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AEStronglyMeasurable.neg
+
+/- warning: measure_theory.ae_strongly_measurable.div -> MeasureTheory.AEStronglyMeasurable.div is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (HDiv.hDiv.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHDiv.{max u1 u2} (α -> β) (Pi.instDiv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toHasDiv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4)))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : Group.{u2} β] [_inst_5 : TopologicalGroup.{u2} β _inst_2 _inst_4], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (HDiv.hDiv.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHDiv.{max u1 u2} (α -> β) (Pi.instDiv.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => DivInvMonoid.toDiv.{u2} β (Group.toDivInvMonoid.{u2} β _inst_4)))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.divₓ'. -/
@[to_additive]
-protected theorem div [Group β] [TopologicalGroup β] (hf : AeStronglyMeasurable f μ)
- (hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (f / g) μ :=
+protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
+ (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ :=
⟨hf.mk f / hg.mk g, hf.stronglyMeasurable_mk.div hg.stronglyMeasurable_mk,
hf.ae_eq_mk.div hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.div MeasureTheory.AeStronglyMeasurable.div
-#align measure_theory.ae_strongly_measurable.sub MeasureTheory.AeStronglyMeasurable.sub
+#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.div
+#align measure_theory.ae_strongly_measurable.sub MeasureTheory.AEStronglyMeasurable.sub
+#print MeasureTheory.AEStronglyMeasurable.smul /-
@[to_additive]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
- {g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
- AeStronglyMeasurable (fun x => f x • g x) μ :=
- continuous_smul.comp_aeStronglyMeasurable (hf.prod_mk hg)
-#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AeStronglyMeasurable.smul
-#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AeStronglyMeasurable.vadd
+ {g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
+ AEStronglyMeasurable (fun x => f x • g x) μ :=
+ continuous_smul.comp_aestronglyMeasurable (hf.prod_mk hg)
+#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AEStronglyMeasurable.smul
+#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AEStronglyMeasurable.vadd
+-/
+#print MeasureTheory.AEStronglyMeasurable.const_smul /-
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
- (hf : AeStronglyMeasurable f μ) (c : 𝕜) : AeStronglyMeasurable (c • f) μ :=
+ (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.stronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
-#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AeStronglyMeasurable.const_smul
+#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AEStronglyMeasurable.const_smul
+-/
+#print MeasureTheory.AEStronglyMeasurable.const_smul' /-
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
- (hf : AeStronglyMeasurable f μ) (c : 𝕜) : AeStronglyMeasurable (fun x => c • f x) μ :=
+ (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ :=
hf.const_smul c
-#align measure_theory.ae_strongly_measurable.const_smul' MeasureTheory.AeStronglyMeasurable.const_smul'
+#align measure_theory.ae_strongly_measurable.const_smul' MeasureTheory.AEStronglyMeasurable.const_smul'
+-/
+#print MeasureTheory.AEStronglyMeasurable.smul_const /-
@[to_additive]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
- (hf : AeStronglyMeasurable f μ) (c : β) : AeStronglyMeasurable (fun x => f x • c) μ :=
- continuous_smul.comp_aeStronglyMeasurable (hf.prod_mk aeStronglyMeasurable_const)
-#align measure_theory.ae_strongly_measurable.smul_const MeasureTheory.AeStronglyMeasurable.smul_const
-#align measure_theory.ae_strongly_measurable.vadd_const MeasureTheory.AeStronglyMeasurable.vadd_const
+ (hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ :=
+ continuous_smul.comp_aestronglyMeasurable (hf.prod_mk aestronglyMeasurable_const)
+#align measure_theory.ae_strongly_measurable.smul_const MeasureTheory.AEStronglyMeasurable.smul_const
+#align measure_theory.ae_strongly_measurable.vadd_const MeasureTheory.AEStronglyMeasurable.vadd_const
+-/
end Arithmetic
section Order
-protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AeStronglyMeasurable f μ)
- (hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (f ⊔ g) μ :=
+/- warning: measure_theory.ae_strongly_measurable.sup -> MeasureTheory.AEStronglyMeasurable.sup is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.supₓ'. -/
+protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
+ (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.stronglyMeasurable_mk.sup hg.stronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AeStronglyMeasurable.sup
-
-protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AeStronglyMeasurable f μ)
- (hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (f ⊓ g) μ :=
+#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.sup
+
+/- warning: measure_theory.ae_strongly_measurable.inf -> MeasureTheory.AEStronglyMeasurable.inf is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.inf MeasureTheory.AEStronglyMeasurable.infₓ'. -/
+protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
+ (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.stronglyMeasurable_mk.inf hg.stronglyMeasurable_mk,
hf.ae_eq_mk.inf hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.inf MeasureTheory.AeStronglyMeasurable.inf
+#align measure_theory.ae_strongly_measurable.inf MeasureTheory.AEStronglyMeasurable.inf
end Order
@@ -1420,23 +2042,35 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
+/- warning: list.ae_strongly_measurable_prod' -> List.aestronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : Monoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))] (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (List.prod.{max u1 u2} (α -> M) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))) l) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : Monoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))] (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (List.prod.{max u2 u1} (α -> M) (Pi.instMul.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))) (Pi.instOne.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => Monoid.toOne.{u1} M _inst_4)) l) μ)
+Case conversion may be inaccurate. Consider using '#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
-theorem List.aeStronglyMeasurable_prod' (l : List (α → M))
- (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) : AeStronglyMeasurable l.Prod μ :=
+theorem List.aestronglyMeasurable_prod' (l : List (α → M))
+ (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
by
induction' l with f l ihl; · exact ae_strongly_measurable_one
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
-#align list.ae_strongly_measurable_prod' List.aeStronglyMeasurable_prod'
-#align list.ae_strongly_measurable_sum' List.aeStronglyMeasurable_sum'
-
+#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
+#align list.ae_strongly_measurable_sum' List.aestronglyMeasurable_sum'
+
+/- warning: list.ae_strongly_measurable_prod -> List.aestronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : Monoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4))] (l : List.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (List.{max u1 u2} (α -> M)) (List.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (x : α) => List.prod.{u2} M (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4)) (MulOneClass.toHasOne.{u2} M (Monoid.toMulOneClass.{u2} M _inst_4)) (List.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) l)) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : Monoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4))] (l : List.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (List.{max u2 u1} (α -> M)) (List.instMembershipList.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (x : α) => List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_4)) (Monoid.toOne.{u1} M _inst_4) (List.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) l)) μ)
+Case conversion may be inaccurate. Consider using '#align list.ae_strongly_measurable_prod List.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
-theorem List.aeStronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) :
- AeStronglyMeasurable (fun x => (l.map fun f : α → M => f x).Prod) μ := by
+theorem List.aestronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
+ AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).Prod) μ := by
simpa only [← Pi.list_prod_apply] using l.ae_strongly_measurable_prod' hl
-#align list.ae_strongly_measurable_prod List.aeStronglyMeasurable_prod
-#align list.ae_strongly_measurable_sum List.aeStronglyMeasurable_sum
+#align list.ae_strongly_measurable_prod List.aestronglyMeasurable_prod
+#align list.ae_strongly_measurable_sum List.aestronglyMeasurable_sum
end Monoid
@@ -1444,39 +2078,63 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
+/- warning: multiset.ae_strongly_measurable_prod' -> Multiset.aestronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] (l : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (Multiset.prod.{max u1 u2} (α -> M) (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) l) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] (l : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f l) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (Multiset.prod.{max u2 u1} (α -> M) (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) l) μ)
+Case conversion may be inaccurate. Consider using '#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
-theorem Multiset.aeStronglyMeasurable_prod' (l : Multiset (α → M))
- (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) : AeStronglyMeasurable l.Prod μ :=
+theorem Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
+ (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.Prod μ :=
by
rcases l with ⟨l⟩
simpa using l.ae_strongly_measurable_prod' (by simpa using hl)
-#align multiset.ae_strongly_measurable_prod' Multiset.aeStronglyMeasurable_prod'
-#align multiset.ae_strongly_measurable_sum' Multiset.aeStronglyMeasurable_sum'
-
+#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'
+#align multiset.ae_strongly_measurable_sum' Multiset.aestronglyMeasurable_sum'
+
+/- warning: multiset.ae_strongly_measurable_prod -> Multiset.aestronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] (s : Multiset.{max u1 u2} (α -> M)), (forall (f : α -> M), (Membership.Mem.{max u1 u2, max u1 u2} (α -> M) (Multiset.{max u1 u2} (α -> M)) (Multiset.hasMem.{max u1 u2} (α -> M)) f s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (x : α) => Multiset.prod.{u2} M _inst_4 (Multiset.map.{max u1 u2, u2} (α -> M) M (fun (f : α -> M) => f x) s)) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] (s : Multiset.{max u2 u1} (α -> M)), (forall (f : α -> M), (Membership.mem.{max u2 u1, max u2 u1} (α -> M) (Multiset.{max u2 u1} (α -> M)) (Multiset.instMembershipMultiset.{max u2 u1} (α -> M)) f s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m f μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (x : α) => Multiset.prod.{u1} M _inst_4 (Multiset.map.{max u2 u1, u1} (α -> M) M (fun (f : α -> M) => f x) s)) μ)
+Case conversion may be inaccurate. Consider using '#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
-theorem Multiset.aeStronglyMeasurable_prod (s : Multiset (α → M))
- (hs : ∀ f ∈ s, AeStronglyMeasurable f μ) :
- AeStronglyMeasurable (fun x => (s.map fun f : α → M => f x).Prod) μ := by
+theorem Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
+ (hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
+ AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).Prod) μ := by
simpa only [← Pi.multiset_prod_apply] using s.ae_strongly_measurable_prod' hs
-#align multiset.ae_strongly_measurable_prod Multiset.aeStronglyMeasurable_prod
-#align multiset.ae_strongly_measurable_sum Multiset.aeStronglyMeasurable_sum
-
+#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prod
+#align multiset.ae_strongly_measurable_sum Multiset.aestronglyMeasurable_sum
+
+/- warning: finset.ae_strongly_measurable_prod' -> Finset.aestronglyMeasurable_prod' is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u1, u2} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) s (fun (i : ι) => f i)) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (Finset.prod.{max u1 u2, u3} (α -> M) ι (Pi.commMonoid.{u2, u1} α (fun (ᾰ : α) => M) (fun (i : α) => _inst_4)) s (fun (i : ι) => f i)) μ)
+Case conversion may be inaccurate. Consider using '#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'ₓ'. -/
@[to_additive]
-theorem Finset.aeStronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
- (hf : ∀ i ∈ s, AeStronglyMeasurable (f i) μ) : AeStronglyMeasurable (∏ i in s, f i) μ :=
- Multiset.aeStronglyMeasurable_prod' _ fun g hg =>
+theorem Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
+ (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i in s, f i) μ :=
+ Multiset.aestronglyMeasurable_prod' _ fun g hg =>
let ⟨i, hi, hg⟩ := Multiset.mem_map.1 hg
hg ▸ hf _ hi
-#align finset.ae_strongly_measurable_prod' Finset.aeStronglyMeasurable_prod'
-#align finset.ae_strongly_measurable_sum' Finset.aeStronglyMeasurable_sum'
-
+#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'
+#align finset.ae_strongly_measurable_sum' Finset.aestronglyMeasurable_sum'
+
+/- warning: finset.ae_strongly_measurable_prod -> Finset.aestronglyMeasurable_prod is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {M : Type.{u2}} [_inst_4 : CommMonoid.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousMul.{u2} M _inst_5 (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M (CommMonoid.toMonoid.{u2} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α M _inst_5 m (fun (a : α) => Finset.prod.{u2, u3} M ι _inst_4 s (fun (i : ι) => f i a)) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {M : Type.{u1}} [_inst_4 : CommMonoid.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousMul.{u1} M _inst_5 (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_4)))] {ι : Type.{u3}} {f : ι -> α -> M} (s : Finset.{u3} ι), (forall (i : ι), (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i s) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (f i) μ)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u1} α M _inst_5 m (fun (a : α) => Finset.prod.{u1, u3} M ι _inst_4 s (fun (i : ι) => f i a)) μ)
+Case conversion may be inaccurate. Consider using '#align finset.ae_strongly_measurable_prod Finset.aestronglyMeasurable_prodₓ'. -/
@[to_additive]
-theorem Finset.aeStronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
- (hf : ∀ i ∈ s, AeStronglyMeasurable (f i) μ) :
- AeStronglyMeasurable (fun a => ∏ i in s, f i a) μ := by
+theorem Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
+ (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
+ AEStronglyMeasurable (fun a => ∏ i in s, f i a) μ := by
simpa only [← Finset.prod_apply] using s.ae_strongly_measurable_prod' hf
-#align finset.ae_strongly_measurable_prod Finset.aeStronglyMeasurable_prod
-#align finset.ae_strongly_measurable_sum Finset.aeStronglyMeasurable_sum
+#align finset.ae_strongly_measurable_prod Finset.aestronglyMeasurable_prod
+#align finset.ae_strongly_measurable_sum Finset.aestronglyMeasurable_sum
end CommMonoid
@@ -1484,60 +2142,87 @@ section SecondCountableAeStronglyMeasurable
variable [MeasurableSpace β]
+#print AEMeasurable.aestronglyMeasurable /-
/-- In a space with second countable topology, measurable implies strongly measurable. -/
-theorem AEMeasurable.aeStronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
- [SecondCountableTopology β] (hf : AEMeasurable f μ) : AeStronglyMeasurable f μ :=
+theorem AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
+ [SecondCountableTopology β] (hf : AEMeasurable f μ) : AEStronglyMeasurable f μ :=
⟨hf.mk f, hf.measurable_mk.StronglyMeasurable, hf.ae_eq_mk⟩
-#align ae_measurable.ae_strongly_measurable AEMeasurable.aeStronglyMeasurable
+#align ae_measurable.ae_strongly_measurable AEMeasurable.aestronglyMeasurable
+-/
-theorem aeStronglyMeasurable_id {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
+#print aestronglyMeasurable_id /-
+theorem aestronglyMeasurable_id {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
{m : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
- AeStronglyMeasurable (id : α → α) μ :=
- aemeasurable_id.AeStronglyMeasurable
-#align ae_strongly_measurable_id aeStronglyMeasurable_id
+ AEStronglyMeasurable (id : α → α) μ :=
+ aemeasurable_id.AEStronglyMeasurable
+#align ae_strongly_measurable_id aestronglyMeasurable_id
+-/
+#print aestronglyMeasurable_iff_aemeasurable /-
/-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
-theorem aeStronglyMeasurable_iff_aEMeasurable [PseudoMetrizableSpace β] [BorelSpace β]
- [SecondCountableTopology β] : AeStronglyMeasurable f μ ↔ AEMeasurable f μ :=
- ⟨fun h => h.AEMeasurable, fun h => h.AeStronglyMeasurable⟩
-#align ae_strongly_measurable_iff_ae_measurable aeStronglyMeasurable_iff_aEMeasurable
+theorem aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β] [BorelSpace β]
+ [SecondCountableTopology β] : AEStronglyMeasurable f μ ↔ AEMeasurable f μ :=
+ ⟨fun h => h.AEMeasurable, fun h => h.AEStronglyMeasurable⟩
+#align ae_strongly_measurable_iff_ae_measurable aestronglyMeasurable_iff_aemeasurable
+-/
end SecondCountableAeStronglyMeasurable
+#print MeasureTheory.AEStronglyMeasurable.dist /-
protected theorem dist {β : Type _} [PseudoMetricSpace β] {f g : α → β}
- (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
- AeStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
- continuous_dist.comp_aeStronglyMeasurable (hf.prod_mk hg)
-#align measure_theory.ae_strongly_measurable.dist MeasureTheory.AeStronglyMeasurable.dist
+ (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
+ AEStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
+ continuous_dist.comp_aestronglyMeasurable (hf.prod_mk hg)
+#align measure_theory.ae_strongly_measurable.dist MeasureTheory.AEStronglyMeasurable.dist
+-/
+#print MeasureTheory.AEStronglyMeasurable.norm /-
protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
- (hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => ‖f x‖) μ :=
- continuous_norm.comp_aeStronglyMeasurable hf
-#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AeStronglyMeasurable.norm
+ (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖) μ :=
+ continuous_norm.comp_aestronglyMeasurable hf
+#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AEStronglyMeasurable.norm
+-/
+/- warning: measure_theory.ae_strongly_measurable.nnnorm -> MeasureTheory.AEStronglyMeasurable.nnnorm is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_4 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_4))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_4)) (f x)) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_4 : SeminormedAddCommGroup.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} β _inst_4))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => NNNorm.nnnorm.{u2} β (SeminormedAddGroup.toNNNorm.{u2} β (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} β _inst_4)) (f x)) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnormₓ'. -/
protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
- (hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => ‖f x‖₊) μ :=
- continuous_nnnorm.comp_aeStronglyMeasurable hf
-#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AeStronglyMeasurable.nnnorm
+ (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ :=
+ continuous_nnnorm.comp_aestronglyMeasurable hf
+#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnorm
+#print MeasureTheory.AEStronglyMeasurable.ennnorm /-
protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
- (hf : AeStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
- (ENNReal.continuous_coe.comp_aeStronglyMeasurable hf.nnnorm).AEMeasurable
-#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AeStronglyMeasurable.ennnorm
+ (hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
+ (ENNReal.continuous_coe.comp_aestronglyMeasurable hf.nnnorm).AEMeasurable
+#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AEStronglyMeasurable.ennnorm
+-/
+#print MeasureTheory.AEStronglyMeasurable.edist /-
protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α → β}
- (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
+ (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEMeasurable (fun a => edist (f a) (g a)) μ :=
- (continuous_edist.comp_aeStronglyMeasurable (hf.prod_mk hg)).AEMeasurable
-#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AeStronglyMeasurable.edist
-
-protected theorem real_toNNReal {f : α → ℝ} (hf : AeStronglyMeasurable f μ) :
- AeStronglyMeasurable (fun x => (f x).toNNReal) μ :=
- continuous_real_toNNReal.comp_aeStronglyMeasurable hf
-#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AeStronglyMeasurable.real_toNNReal
+ (continuous_edist.comp_aestronglyMeasurable (hf.prod_mk hg)).AEMeasurable
+#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AEStronglyMeasurable.edist
+-/
-theorem aeStronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
- AeStronglyMeasurable (indicator s f) μ ↔ AeStronglyMeasurable f (μ.restrict s) :=
+/- warning: measure_theory.ae_strongly_measurable.real_to_nnreal -> MeasureTheory.AEStronglyMeasurable.real_toNNReal is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (MeasureTheory.AEStronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.topologicalSpace m (fun (x : α) => Real.toNNReal (f x)) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (MeasureTheory.AEStronglyMeasurable.{u1, 0} α Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, 0} α NNReal NNReal.instTopologicalSpaceNNReal m (fun (x : α) => Real.toNNReal (f x)) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AEStronglyMeasurable.real_toNNRealₓ'. -/
+protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
+ AEStronglyMeasurable (fun x => (f x).toNNReal) μ :=
+ continuous_real_toNNReal.comp_aestronglyMeasurable hf
+#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AEStronglyMeasurable.real_toNNReal
+
+#print aestronglyMeasurable_indicator_iff /-
+theorem aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
+ AEStronglyMeasurable (indicator s f) μ ↔ AEStronglyMeasurable f (μ.restrict s) :=
by
constructor
· intro h
@@ -1549,15 +2234,19 @@ theorem aeStronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : Measurab
have B : s.indicator f =ᵐ[μ.restrict (sᶜ)] s.indicator (h.mk f) :=
(indicator_ae_eq_restrict_compl hs).trans (indicator_ae_eq_restrict_compl hs).symm
exact ae_of_ae_restrict_of_ae_restrict_compl _ A B
-#align ae_strongly_measurable_indicator_iff aeStronglyMeasurable_indicator_iff
+#align ae_strongly_measurable_indicator_iff aestronglyMeasurable_indicator_iff
+-/
-protected theorem indicator [Zero β] (hfm : AeStronglyMeasurable f μ) {s : Set α}
- (hs : MeasurableSet s) : AeStronglyMeasurable (s.indicator f) μ :=
- (aeStronglyMeasurable_indicator_iff hs).mpr hfm.restrict
-#align measure_theory.ae_strongly_measurable.indicator MeasureTheory.AeStronglyMeasurable.indicator
+#print MeasureTheory.AEStronglyMeasurable.indicator /-
+protected theorem indicator [Zero β] (hfm : AEStronglyMeasurable f μ) {s : Set α}
+ (hs : MeasurableSet s) : AEStronglyMeasurable (s.indicator f) μ :=
+ (aestronglyMeasurable_indicator_iff hs).mpr hfm.restrict
+#align measure_theory.ae_strongly_measurable.indicator MeasureTheory.AEStronglyMeasurable.indicator
+-/
+#print MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun /-
theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
- (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
+ (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { x | f x = g x } μ :=
by
apply
@@ -1566,10 +2255,17 @@ theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x = hg.mk g x) = (f x = g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_eq_fun
+#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
+-/
+/- warning: measure_theory.ae_strongly_measurable.null_measurable_set_lt -> MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : LinearOrder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_4))))] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_4))))) (f a) (g a))) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : LinearOrder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_4)))))] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_4)))))) (f a) (g a))) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_ltₓ'. -/
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
- {f g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
+ {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a < g a } μ :=
by
apply
@@ -1577,10 +2273,16 @@ theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMe
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x < hg.mk g x) = (f x < g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_lt
-
+#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
+
+/- warning: measure_theory.ae_strongly_measurable.null_measurable_set_le -> MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : Preorder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 _inst_4] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LE.le.{u2} β (Preorder.toHasLe.{u2} β _inst_4) (f a) (g a))) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : Preorder.{u2} β] [_inst_5 : OrderClosedTopology.{u2} β _inst_2 _inst_4] [_inst_6 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {g : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m g μ) -> (MeasureTheory.NullMeasurableSet.{u1} α m (setOf.{u1} α (fun (a : α) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_4) (f a) (g a))) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_leₓ'. -/
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
- {f g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
+ {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ :=
by
apply
@@ -1588,46 +2290,73 @@ theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetri
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_le
+#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le
-theorem aeStronglyMeasurable_of_aeStronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
- {μ : Measure α} (hm : m ≤ m0) {f : α → β} (hf : AeStronglyMeasurable f (μ.trim hm)) :
- AeStronglyMeasurable f μ :=
+#print aestronglyMeasurable_of_aestronglyMeasurable_trim /-
+theorem aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
+ {μ : Measure α} (hm : m ≤ m0) {f : α → β} (hf : AEStronglyMeasurable f (μ.trim hm)) :
+ AEStronglyMeasurable f μ :=
⟨hf.mk f, StronglyMeasurable.mono hf.stronglyMeasurable_mk hm, ae_eq_of_ae_eq_trim hf.ae_eq_mk⟩
-#align ae_strongly_measurable_of_ae_strongly_measurable_trim aeStronglyMeasurable_of_aeStronglyMeasurable_trim
+#align ae_strongly_measurable_of_ae_strongly_measurable_trim aestronglyMeasurable_of_aestronglyMeasurable_trim
+-/
-theorem comp_aEMeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
- {μ : Measure γ} (hg : AeStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
- AeStronglyMeasurable (g ∘ f) μ :=
+/- warning: measure_theory.ae_strongly_measurable.comp_ae_measurable -> MeasureTheory.AEStronglyMeasurable.comp_aemeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) -> (AEMeasurable.{u3, u1} γ α mα mγ f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) -> (AEMeasurable.{u3, u2} γ α mα mγ f μ) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurableₓ'. -/
+theorem comp_aemeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
+ {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
+ AEStronglyMeasurable (g ∘ f) μ :=
⟨hg.mk g ∘ hf.mk f, hg.stronglyMeasurable_mk.comp_measurable hf.measurable_mk,
(ae_eq_comp hf hg.ae_eq_mk).trans (hf.ae_eq_mk.fun_comp (hg.mk g))⟩
-#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AeStronglyMeasurable.comp_aEMeasurable
-
+#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
+
+/- warning: measure_theory.ae_strongly_measurable.comp_measurable -> MeasureTheory.AEStronglyMeasurable.comp_measurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) -> (Measurable.{u3, u1} γ α mγ mα f) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) -> (Measurable.{u3, u2} γ α mγ mα f) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurableₓ'. -/
theorem comp_measurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
- {μ : Measure γ} (hg : AeStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
- AeStronglyMeasurable (g ∘ f) μ :=
+ {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
+ AEStronglyMeasurable (g ∘ f) μ :=
hg.comp_aemeasurable hf.AEMeasurable
-#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AeStronglyMeasurable.comp_measurable
-
+#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurable
+
+/- warning: measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving -> MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ} {ν : MeasureTheory.Measure.{u1} α mα}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g ν) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u3, u1} γ α mα mγ f μ ν) -> (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {g : α -> β} {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ} {ν : MeasureTheory.Measure.{u2} α mα}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g ν) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u3, u2} γ α mα mγ f μ ν) -> (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreservingₓ'. -/
theorem comp_quasiMeasurePreserving {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α}
- {f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AeStronglyMeasurable g ν)
- (hf : QuasiMeasurePreserving f μ ν) : AeStronglyMeasurable (g ∘ f) μ :=
+ {f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
+ (hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
(hg.mono' hf.AbsolutelyContinuous).comp_measurable hf.Measurable
-#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AeStronglyMeasurable.comp_quasiMeasurePreserving
-
-theorem isSeparable_ae_range (hf : AeStronglyMeasurable f μ) :
+#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
+
+/- warning: measure_theory.ae_strongly_measurable.is_separable_ae_range -> MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (TopologicalSpace.IsSeparable.{u2} β _inst_2 t) (Filter.Eventually.{u1} α (fun (x : α) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (f x) t) (MeasureTheory.Measure.ae.{u1} α m μ))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (Exists.{succ u1} (Set.{u1} β) (fun (t : Set.{u1} β) => And (TopologicalSpace.IsSeparable.{u1} β _inst_2 t) (Filter.Eventually.{u2} α (fun (x : α) => Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (f x) t) (MeasureTheory.Measure.ae.{u2} α m μ))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AEStronglyMeasurable.isSeparable_ae_rangeₓ'. -/
+theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
refine' ⟨range (hf.mk f), hf.strongly_measurable_mk.is_separable_range, _⟩
filter_upwards [hf.ae_eq_mk]with x hx
simp [hx]
-#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AeStronglyMeasurable.isSeparable_ae_range
+#align measure_theory.ae_strongly_measurable.is_separable_ae_range MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
+#print aestronglyMeasurable_iff_aemeasurable_separable /-
/-- A function is almost everywhere strongly measurable if and only if it is almost everywhere
measurable, and up to a zero measure set its range is contained in a separable set. -/
-theorem aeStronglyMeasurable_iff_aEMeasurable_separable [PseudoMetrizableSpace β]
+theorem aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] :
- AeStronglyMeasurable f μ ↔ AEMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
+ AEStronglyMeasurable f μ ↔ AEMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
refine' ⟨fun H => ⟨H.AEMeasurable, H.isSeparable_ae_range⟩, _⟩
rintro ⟨H, ⟨t, t_sep, ht⟩⟩
@@ -1639,26 +2368,39 @@ theorem aeStronglyMeasurable_iff_aEMeasurable_separable [PseudoMetrizableSpace
H.exists_ae_eq_range_subset ht h₀
refine' ⟨g, _, fg⟩
exact stronglyMeasurable_iff_measurable_separable.2 ⟨g_meas, t_sep.mono GT.gt⟩
-#align ae_strongly_measurable_iff_ae_measurable_separable aeStronglyMeasurable_iff_aEMeasurable_separable
+#align ae_strongly_measurable_iff_ae_measurable_separable aestronglyMeasurable_iff_aemeasurable_separable
+-/
-theorem MeasurableEmbedding.aeStronglyMeasurable_map_iff {γ : Type _} {mγ : MeasurableSpace γ}
+/- warning: measurable_embedding.ae_strongly_measurable_map_iff -> MeasurableEmbedding.aestronglyMeasurable_map_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u1} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasurableEmbedding.{u3, u1} γ α mγ mα f) -> (forall {g : α -> β}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u1} γ α mα mγ f μ)) (MeasureTheory.AEStronglyMeasurable.{u3, u2} γ β _inst_2 mγ (Function.comp.{succ u3, succ u1, succ u2} γ α β g f) μ))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {γ : Type.{u3}} {mγ : MeasurableSpace.{u3} γ} {mα : MeasurableSpace.{u2} α} {f : γ -> α} {μ : MeasureTheory.Measure.{u3} γ mγ}, (MeasurableEmbedding.{u3, u2} γ α mγ mα f) -> (forall {g : α -> β}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 mα g (MeasureTheory.Measure.map.{u3, u2} γ α mα mγ f μ)) (MeasureTheory.AEStronglyMeasurable.{u3, u1} γ β _inst_2 mγ (Function.comp.{succ u3, succ u2, succ u1} γ α β g f) μ))
+Case conversion may be inaccurate. Consider using '#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aestronglyMeasurable_map_iffₓ'. -/
+theorem MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _} {mγ : MeasurableSpace γ}
{mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ} (hf : MeasurableEmbedding f) {g : α → β} :
- AeStronglyMeasurable g (Measure.map f μ) ↔ AeStronglyMeasurable (g ∘ f) μ :=
+ AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ :=
by
refine' ⟨fun H => H.comp_measurable hf.measurable, _⟩
rintro ⟨g₁, hgm₁, heq⟩
rcases hf.exists_strongly_measurable_extend hgm₁ fun x => ⟨g x⟩ with ⟨g₂, hgm₂, rfl⟩
exact ⟨g₂, hgm₂, hf.ae_map_iff.2 HEq⟩
-#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aeStronglyMeasurable_map_iff
-
-theorem Embedding.aeStronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [PseudoMetrizableSpace γ]
+#align measurable_embedding.ae_strongly_measurable_map_iff MeasurableEmbedding.aestronglyMeasurable_map_iff
+
+/- warning: embedding.ae_strongly_measurable_comp_iff -> Embedding.aestronglyMeasurable_comp_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u3} γ _inst_3] {g : β -> γ} {f : α -> β}, (Embedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u3} α γ _inst_3 m (fun (x : α) => g (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ))
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} γ _inst_3] {g : β -> γ} {f : α -> β}, (Embedding.{u3, u2} β γ _inst_2 _inst_3 g) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 m (fun (x : α) => g (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u3} α β _inst_2 m f μ))
+Case conversion may be inaccurate. Consider using '#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iffₓ'. -/
+theorem Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [PseudoMetrizableSpace γ]
{g : β → γ} {f : α → β} (hg : Embedding g) :
- AeStronglyMeasurable (fun x => g (f x)) μ ↔ AeStronglyMeasurable f μ :=
+ AEStronglyMeasurable (fun x => g (f x)) μ ↔ AEStronglyMeasurable f μ :=
by
letI := pseudo_metrizable_space_pseudo_metric γ
borelize β γ
refine'
- ⟨fun H => aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩, fun H =>
+ ⟨fun H => aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨_, _⟩, fun H =>
hg.continuous.comp_ae_strongly_measurable H⟩
· let G : β → range g := cod_restrict g (range g) mem_range_self
have hG : ClosedEmbedding G :=
@@ -1670,29 +2412,36 @@ theorem Embedding.aeStronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
exact mem_range_self x }
have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.ae_measurable
exact hG.measurable_embedding.ae_measurable_comp_iff.1 this
- · rcases(aeStronglyMeasurable_iff_aEMeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
+ · rcases(aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
exact ⟨g ⁻¹' t, hg.is_separable_preimage ht, h't⟩
-#align embedding.ae_strongly_measurable_comp_iff Embedding.aeStronglyMeasurable_comp_iff
-
-theorem MeasureTheory.MeasurePreserving.aeStronglyMeasurable_comp_iff {β : Type _} {f : α → β}
+#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iff
+
+/- warning: measure_theory.measure_preserving.ae_strongly_measurable_comp_iff -> MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {γ : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} γ] {β : Type.{u3}} {f : α -> β} {mα : MeasurableSpace.{u1} α} {μa : MeasureTheory.Measure.{u1} α mα} {mβ : MeasurableSpace.{u3} β} {μb : MeasureTheory.Measure.{u3} β mβ}, (MeasureTheory.MeasurePreserving.{u1, u3} α β mα mβ f μa μb) -> (MeasurableEmbedding.{u1, u3} α β mα mβ f) -> (forall {g : β -> γ}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α γ _inst_3 mα (Function.comp.{succ u1, succ u3, succ u2} α β γ g f) μa) (MeasureTheory.AEStronglyMeasurable.{u3, u2} β γ _inst_3 mβ g μb))
+but is expected to have type
+ forall {α : Type.{u2}} {γ : Type.{u1}} [_inst_3 : TopologicalSpace.{u1} γ] {β : Type.{u3}} {f : α -> β} {mα : MeasurableSpace.{u2} α} {μa : MeasureTheory.Measure.{u2} α mα} {mβ : MeasurableSpace.{u3} β} {μb : MeasureTheory.Measure.{u3} β mβ}, (MeasureTheory.MeasurePreserving.{u2, u3} α β mα mβ f μa μb) -> (MeasurableEmbedding.{u2, u3} α β mα mβ f) -> (forall {g : β -> γ}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α γ _inst_3 mα (Function.comp.{succ u2, succ u3, succ u1} α β γ g f) μa) (MeasureTheory.AEStronglyMeasurable.{u3, u1} β γ _inst_3 mβ g μb))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure_preserving.ae_strongly_measurable_comp_iff MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iffₓ'. -/
+theorem MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type _} {f : α → β}
{mα : MeasurableSpace α} {μa : Measure α} {mβ : MeasurableSpace β} {μb : Measure β}
(hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ} :
- AeStronglyMeasurable (g ∘ f) μa ↔ AeStronglyMeasurable g μb := by
+ AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb := by
rw [← hf.map_eq, h₂.ae_strongly_measurable_map_iff]
-#align measure_theory.measure_preserving.ae_strongly_measurable_comp_iff MeasureTheory.MeasurePreserving.aeStronglyMeasurable_comp_iff
+#align measure_theory.measure_preserving.ae_strongly_measurable_comp_iff MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff
+#print aestronglyMeasurable_of_tendsto_ae /-
/-- An almost everywhere sequential limit of almost everywhere strongly measurable functions is
almost everywhere strongly measurable. -/
-theorem aeStronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace β] (u : Filter ι)
+theorem aestronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace β] (u : Filter ι)
[NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
- (hf : ∀ i, AeStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
- AeStronglyMeasurable g μ := by
+ (hf : ∀ i, AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
+ AEStronglyMeasurable g μ := by
borelize β
- refine' aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩
+ refine' aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨_, _⟩
· exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).AEMeasurable) limUnder
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
- (aeStronglyMeasurable_iff_aEMeasurable_separable.1 (hf (v n))).2
+ (aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
choose t t_sep ht using this
refine' ⟨closure (⋃ i, t i), (is_separable_Union fun i => t_sep i).closure, _⟩
filter_upwards [ae_all_iff.2 ht, limUnder]with x hx h'x
@@ -1700,12 +2449,14 @@ theorem aeStronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace
apply eventually_of_forall fun n => _
apply mem_Union_of_mem n
exact hx n
-#align ae_strongly_measurable_of_tendsto_ae aeStronglyMeasurable_of_tendsto_ae
+#align ae_strongly_measurable_of_tendsto_ae aestronglyMeasurable_of_tendsto_ae
+-/
+#print exists_stronglyMeasurable_limit_of_tendsto_ae /-
/-- If a sequence of almost everywhere strongly measurable functions converges almost everywhere,
one can select a strongly measurable function as the almost everywhere limit. -/
theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β] {f : ℕ → α → β}
- (hf : ∀ n, AeStronglyMeasurable (f n) μ)
+ (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
∃ (f_lim : α → β)(hf_lim_meas : StronglyMeasurable f_lim),
∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) :=
@@ -1714,82 +2465,125 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
obtain ⟨g, g_meas, hg⟩ :
∃ (g : α → β)(g_meas : Measurable g), ∀ᵐ x ∂μ, tendsto (fun n => f n x) at_top (𝓝 (g x)) :=
measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).AEMeasurable) h_ae_tendsto
- have Hg : ae_strongly_measurable g μ := aeStronglyMeasurable_of_tendsto_ae _ hf hg
+ have Hg : ae_strongly_measurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.strongly_measurable_mk, _⟩
filter_upwards [hg, Hg.ae_eq_mk]with x hx h'x
rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
+-/
+/- warning: measure_theory.ae_strongly_measurable.sum_measure -> MeasureTheory.AEStronglyMeasurable.sum_measure is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {m : MeasurableSpace.{u1} α} {μ : ι -> (MeasureTheory.Measure.{u1} α m)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (μ i)) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.sum.{u1, u3} α ι m μ))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {m : MeasurableSpace.{u2} α} {μ : ι -> (MeasureTheory.Measure.{u2} α m)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (μ i)) -> (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.sum.{u2, u1} α ι m μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measureₓ'. -/
theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
- (h : ∀ i, AeStronglyMeasurable f (μ i)) : AeStronglyMeasurable f (Measure.sum μ) :=
+ (h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) :=
by
borelize β
refine'
- aeStronglyMeasurable_iff_aEMeasurable_separable.2
+ aestronglyMeasurable_iff_aemeasurable_separable.2
⟨AEMeasurable.sum_measure fun i => (h i).AEMeasurable, _⟩
have A : ∀ i : ι, ∃ t : Set β, IsSeparable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
- (aeStronglyMeasurable_iff_aEMeasurable_separable.1 (h i)).2
+ (aestronglyMeasurable_iff_aemeasurable_separable.1 (h i)).2
choose t t_sep ht using A
refine' ⟨⋃ i, t i, is_separable_Union t_sep, _⟩
simp only [measure.ae_sum_eq, mem_Union, eventually_supr]
intro i
filter_upwards [ht i]with x hx
exact ⟨i, hx⟩
-#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AeStronglyMeasurable.sum_measure
-
+#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measure
+
+/- warning: ae_strongly_measurable_sum_measure_iff -> aestronglyMeasurable_sum_measure_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {m : MeasurableSpace.{u1} α} {μ : ι -> (MeasureTheory.Measure.{u1} α m)}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.sum.{u1, u3} α ι m μ)) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (μ i))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {m : MeasurableSpace.{u2} α} {μ : ι -> (MeasureTheory.Measure.{u2} α m)}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.sum.{u2, u1} α ι m μ)) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (μ i))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_sum_measure_iff aestronglyMeasurable_sum_measure_iffₓ'. -/
@[simp]
-theorem aeStronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : MeasurableSpace α}
- {μ : ι → Measure α} : AeStronglyMeasurable f (Sum μ) ↔ ∀ i, AeStronglyMeasurable f (μ i) :=
+theorem aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : MeasurableSpace α}
+ {μ : ι → Measure α} : AEStronglyMeasurable f (Sum μ) ↔ ∀ i, AEStronglyMeasurable f (μ i) :=
⟨fun h i => h.mono_measure (Measure.le_sum _ _), sum_measure⟩
-#align ae_strongly_measurable_sum_measure_iff aeStronglyMeasurable_sum_measure_iff
+#align ae_strongly_measurable_sum_measure_iff aestronglyMeasurable_sum_measure_iff
+#print aestronglyMeasurable_add_measure_iff /-
@[simp]
-theorem aeStronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
- AeStronglyMeasurable f (μ + ν) ↔ AeStronglyMeasurable f μ ∧ AeStronglyMeasurable f ν :=
+theorem aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
+ AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν :=
by
- rw [← sum_cond, aeStronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]
+ rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]
rfl
-#align ae_strongly_measurable_add_measure_iff aeStronglyMeasurable_add_measure_iff
+#align ae_strongly_measurable_add_measure_iff aestronglyMeasurable_add_measure_iff
+-/
+#print MeasureTheory.AEStronglyMeasurable.add_measure /-
theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
- (hμ : AeStronglyMeasurable f μ) (hν : AeStronglyMeasurable f ν) :
- AeStronglyMeasurable f (μ + ν) :=
- aeStronglyMeasurable_add_measure_iff.2 ⟨hμ, hν⟩
-#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AeStronglyMeasurable.add_measure
+ (hμ : AEStronglyMeasurable f μ) (hν : AEStronglyMeasurable f ν) :
+ AEStronglyMeasurable f (μ + ν) :=
+ aestronglyMeasurable_add_measure_iff.2 ⟨hμ, hν⟩
+#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AEStronglyMeasurable.add_measure
+-/
+/- warning: measure_theory.ae_strongly_measurable.Union -> MeasureTheory.AEStronglyMeasurable.iUnion is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {s : ι -> (Set.{u1} α)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (s i))) -> (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u3} α ι (fun (i : ι) => s i))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {s : ι -> (Set.{u2} α)}, (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (s i))) -> (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AEStronglyMeasurable.iUnionₓ'. -/
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
- (h : ∀ i, AeStronglyMeasurable f (μ.restrict (s i))) :
- AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
+ (h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
+ AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
(sum_measure h).mono_measure <| restrict_iUnion_le
-#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AeStronglyMeasurable.iUnion
-
+#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AEStronglyMeasurable.iUnion
+
+/- warning: ae_strongly_measurable_Union_iff -> aestronglyMeasurable_iUnion_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_1 : Countable.{succ u3} ι] {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {s : ι -> (Set.{u1} α)}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u3} α ι (fun (i : ι) => s i)))) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (s i)))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u3}} {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u3} β] {f : α -> β} [_inst_4 : TopologicalSpace.PseudoMetrizableSpace.{u3} β _inst_2] {s : ι -> (Set.{u2} α)}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i)))) (forall (i : ι), MeasureTheory.AEStronglyMeasurable.{u2, u3} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (s i)))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_Union_iff aestronglyMeasurable_iUnion_iffₓ'. -/
@[simp]
-theorem aeStronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
- AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
- ∀ i, AeStronglyMeasurable f (μ.restrict (s i)) :=
+theorem aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
+ AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
+ ∀ i, AEStronglyMeasurable f (μ.restrict (s i)) :=
⟨fun h i => h.mono_measure <| restrict_mono (subset_iUnion _ _) le_rfl,
- AeStronglyMeasurable.iUnion⟩
-#align ae_strongly_measurable_Union_iff aeStronglyMeasurable_iUnion_iff
+ AEStronglyMeasurable.iUnion⟩
+#align ae_strongly_measurable_Union_iff aestronglyMeasurable_iUnion_iff
+#print aestronglyMeasurable_union_iff /-
@[simp]
-theorem aeStronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α} :
- AeStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
- AeStronglyMeasurable f (μ.restrict s) ∧ AeStronglyMeasurable f (μ.restrict t) :=
- by simp only [union_eq_Union, aeStronglyMeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]
-#align ae_strongly_measurable_union_iff aeStronglyMeasurable_union_iff
+theorem aestronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α} :
+ AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
+ AEStronglyMeasurable f (μ.restrict s) ∧ AEStronglyMeasurable f (μ.restrict t) :=
+ by simp only [union_eq_Union, aestronglyMeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]
+#align ae_strongly_measurable_union_iff aestronglyMeasurable_union_iff
+-/
-theorem aeStronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
+/- warning: measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff -> MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] [_inst_4 : LinearOrder.{u1} α] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_2] {f : α -> β} {a : α} {b : α}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.uIoc.{u1} α _inst_4 a b))) (And (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.Ioc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_4)))) a b))) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u1} α m μ (Set.Ioc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_4)))) b a))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] [_inst_4 : LinearOrder.{u2} α] [_inst_5 : TopologicalSpace.PseudoMetrizableSpace.{u1} β _inst_2] {f : α -> β} {a : α} {b : α}, Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.uIoc.{u2} α _inst_4 a b))) (And (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.Ioc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_4))))) a b))) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (MeasureTheory.Measure.restrict.{u2} α m μ (Set.Ioc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_4))))) b a))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iffₓ'. -/
+theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
{a b : α} :
- AeStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
- AeStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
- AeStronglyMeasurable f (μ.restrict <| Ioc b a) :=
- by rw [uIoc_eq_union, aeStronglyMeasurable_union_iff]
-#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AeStronglyMeasurable.aeStronglyMeasurable_uIoc_iff
-
+ AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
+ AEStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
+ AEStronglyMeasurable f (μ.restrict <| Ioc b a) :=
+ by rw [uIoc_eq_union, aestronglyMeasurable_union_iff]
+#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
+
+/- warning: measure_theory.ae_strongly_measurable.smul_measure -> MeasureTheory.AEStronglyMeasurable.smul_measure is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {R : Type.{u3}} [_inst_4 : Monoid.{u3} R] [_inst_5 : DistribMulAction.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))] [_inst_6 : IsScalarTower.{u3, 0, 0} R ENNReal ENNReal (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5))) (Mul.toSMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5)))], (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ) -> (forall (c : R), MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f (SMul.smul.{u3, u1} R (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, u3} α R (SMulZeroClass.toHasSmul.{u3, 0} R ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)))) (DistribSMul.toSmulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal ENNReal.addCommMonoidWithOne)) _inst_5))) _inst_6 m) c μ))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {R : Type.{u3}} [_inst_4 : Monoid.{u3} R] [_inst_5 : DistribMulAction.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))] [_inst_6 : IsScalarTower.{u3, 0, 0} R ENNReal ENNReal (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5))) (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5)))], (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ) -> (forall (c : R), MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f (HSMul.hSMul.{u3, u2, u2} R (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.{u2} α m) (instHSMul.{u3, u2} R (MeasureTheory.Measure.{u2} α m) (MeasureTheory.Measure.instSMul.{u2, u3} α R (SMulZeroClass.toSMul.{u3, 0} R ENNReal instENNRealZero (DistribSMul.toSMulZeroClass.{u3, 0} R ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne))) (DistribMulAction.toDistribSMul.{u3, 0} R ENNReal _inst_4 (AddMonoidWithOne.toAddMonoid.{0} ENNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal instENNRealAddCommMonoidWithOne)) _inst_5))) _inst_6 m)) c μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AEStronglyMeasurable.smul_measureₓ'. -/
theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
- (h : AeStronglyMeasurable f μ) (c : R) : AeStronglyMeasurable f (c • μ) :=
+ (h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
-#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AeStronglyMeasurable.smul_measure
+#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AEStronglyMeasurable.smul_measure
section NormedSpace
@@ -1797,10 +2591,16 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-theorem aeStronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
- AeStronglyMeasurable (fun x => f x • c) μ ↔ AeStronglyMeasurable f μ :=
- (closedEmbedding_smul_left hc).toEmbedding.aeStronglyMeasurable_comp_iff
-#align ae_strongly_measurable_smul_const_iff aeStronglyMeasurable_smul_const_iff
+/- warning: ae_strongly_measurable_smul_const_iff -> aestronglyMeasurable_smul_const_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] [_inst_5 : CompleteSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (OfNat.mk.{u3} E 0 (Zero.zero.{u3} E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (SubNegMonoid.toAddMonoid.{u3} E (AddGroup.toSubNegMonoid.{u3} E (NormedAddGroup.toAddGroup.{u3} E (NormedAddCommGroup.toNormedAddGroup.{u3} E _inst_6)))))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) m (fun (x : α) => SMul.smul.{u2, u3} 𝕜 E (SMulZeroClass.toHasSmul.{u2, u3} 𝕜 E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} 𝕜 E (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u2, u3} 𝕜 E (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7))))) (f x) c) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) m f μ))
+but is expected to have type
+ forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {𝕜 : Type.{u1}} [_inst_4 : NontriviallyNormedField.{u1} 𝕜] [_inst_5 : CompleteSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))))] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (Zero.toOfNat0.{u3} E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6))))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) m (fun (x : α) => HSMul.hSMul.{u1, u3, u3} 𝕜 E E (instHSMul.{u1, u3} 𝕜 E (SMulZeroClass.toSMul.{u1, u3} 𝕜 E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSMulZeroClass.{u1, u3} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u1, u3} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7)))))) (f x) c) μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α 𝕜 (UniformSpace.toTopologicalSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_4))))))) m f μ))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_smul_const_iff aestronglyMeasurable_smul_const_iffₓ'. -/
+theorem aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
+ AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ :=
+ (closedEmbedding_smul_left hc).toEmbedding.aestronglyMeasurable_comp_iff
+#align ae_strongly_measurable_smul_const_iff aestronglyMeasurable_smul_const_iff
end NormedSpace
@@ -1808,20 +2608,32 @@ section MulAction
variable {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
-theorem aeStronglyMeasurable_const_smul_iff (c : G) :
- AeStronglyMeasurable (fun x => c • f x) μ ↔ AeStronglyMeasurable f μ :=
+/- warning: ae_strongly_measurable_const_smul_iff -> aestronglyMeasurable_const_smul_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u3}} [_inst_4 : Group.{u3} G] [_inst_5 : MulAction.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4))] [_inst_6 : ContinuousConstSMul.{u3, u2} G β _inst_2 (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4)) _inst_5)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G β (MulAction.toHasSmul.{u3, u2} G β (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_4)) _inst_5) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ)
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G : Type.{u1}} [_inst_4 : Group.{u1} G] [_inst_5 : MulAction.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4))] [_inst_6 : ContinuousConstSMul.{u1, u2} G β _inst_2 (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4)) _inst_5)] (c : G), Iff (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u1, u2, u2} G β β (instHSMul.{u1, u2} G β (MulAction.toSMul.{u1, u2} G β (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_4)) _inst_5)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u3, u2} α β _inst_2 m f μ)
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iffₓ'. -/
+theorem aestronglyMeasurable_const_smul_iff (c : G) :
+ AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
-#align ae_strongly_measurable_const_smul_iff aeStronglyMeasurable_const_smul_iff
+#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iff
variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
-theorem aeStronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
- AeStronglyMeasurable (fun x => c • f x) μ ↔ AeStronglyMeasurable f μ :=
+/- warning: ae_strongly_measurable_const_smul_iff₀ -> aestronglyMeasurable_const_smul_iff₀ is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_7 : GroupWithZero.{u3} G₀] [_inst_8 : MulAction.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))] [_inst_9 : ContinuousConstSMul.{u3, u2} G₀ β _inst_2 (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (OfNat.mk.{u3} G₀ 0 (Zero.zero.{u3} G₀ (MulZeroClass.toHasZero.{u3} G₀ (MulZeroOneClass.toMulZeroClass.{u3} G₀ (MonoidWithZero.toMulZeroOneClass.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)))))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m (fun (x : α) => SMul.smul.{u3, u2} G₀ β (MulAction.toHasSmul.{u3, u2} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α β _inst_2 m f μ))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {G₀ : Type.{u3}} [_inst_7 : GroupWithZero.{u3} G₀] [_inst_8 : MulAction.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))] [_inst_9 : ContinuousConstSMul.{u3, u1} G₀ β _inst_2 (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)] {c : G₀}, (Ne.{succ u3} G₀ c (OfNat.ofNat.{u3} G₀ 0 (Zero.toOfNat0.{u3} G₀ (MonoidWithZero.toZero.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7))))) -> (Iff (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m (fun (x : α) => HSMul.hSMul.{u3, u1, u1} G₀ β β (instHSMul.{u3, u1} G₀ β (MulAction.toSMul.{u3, u1} G₀ β (MonoidWithZero.toMonoid.{u3} G₀ (GroupWithZero.toMonoidWithZero.{u3} G₀ _inst_7)) _inst_8)) c (f x)) μ) (MeasureTheory.AEStronglyMeasurable.{u2, u1} α β _inst_2 m f μ))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀ₓ'. -/
+theorem aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
+ AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
by
refine' ⟨fun h => _, fun h => h.const_smul c⟩
convert h.const_smul' c⁻¹
simp [smul_smul, inv_mul_cancel hc]
-#align ae_strongly_measurable_const_smul_iff₀ aeStronglyMeasurable_const_smul_iff₀
+#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀
end MulAction
@@ -1835,28 +2647,52 @@ variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+/- warning: strongly_measurable.apply_continuous_linear_map -> StronglyMeasurable.apply_continuousLinearMap is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {m : MeasurableSpace.{u1} α} {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ) -> (forall (v : F), MeasureTheory.StronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v))
+but is expected to have type
+ forall {α : Type.{u4}} {𝕜 : Type.{u3}} [_inst_4 : NontriviallyNormedField.{u3} 𝕜] {E : Type.{u1}} [_inst_5 : NormedAddCommGroup.{u1} E] [_inst_6 : NormedSpace.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {m : MeasurableSpace.{u4} α} {φ : α -> (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6))}, (MeasureTheory.StronglyMeasurable.{u4, max u1 u2} α (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u3, u3, u2, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E 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(PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 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F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)) (NormedSpace.toModule.{u3, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u3, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u3} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)))) (φ a) v))
+Case conversion may be inaccurate. Consider using '#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMapₓ'. -/
theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {φ : α → F →L[𝕜] E}
(hφ : StronglyMeasurable φ) (v : F) : StronglyMeasurable fun a => φ a v :=
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_stronglyMeasurable hφ
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
-theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AeStronglyMeasurable φ μ) (v : F) :
- AeStronglyMeasurable (fun a => φ a v) μ :=
- (ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_aeStronglyMeasurable hφ
-#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AeStronglyMeasurable.apply_continuousLinearMap
-
-theorem ContinuousLinearMap.aeStronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
- {g : α → F} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
- AeStronglyMeasurable (fun x => L (f x) (g x)) μ :=
- L.continuous₂.comp_aeStronglyMeasurable <| hf.prod_mk hg
-#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aeStronglyMeasurable_comp₂
+/- warning: measure_theory.ae_strongly_measurable.apply_continuous_linear_map -> MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F E (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6)) (φ a) v) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u4}} [_inst_4 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_5 : NormedAddCommGroup.{u2} E] [_inst_6 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6))}, (MeasureTheory.AEStronglyMeasurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (ContinuousLinearMap.topologicalSpace.{u4, u4, u3, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F E (SeminormedAddCommGroup.toAddCommGroup.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5))) m φ μ) -> (forall (v : F), MeasureTheory.AEStronglyMeasurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (UniformSpace.toTopologicalSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (PseudoMetricSpace.toUniformSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5)))) m (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_5)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)))) (φ a) v) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMapₓ'. -/
+theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
+ AEStronglyMeasurable (fun a => φ a v) μ :=
+ (ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_aestronglyMeasurable hφ
+#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
+
+/- warning: continuous_linear_map.ae_strongly_measurable_comp₂ -> ContinuousLinearMap.aestronglyMeasurable_comp₂ is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u2}} [_inst_4 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} E] [_inst_6 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {G : Type.{u5}} [_inst_9 : NormedAddCommGroup.{u5} G] [_inst_10 : NormedSpace.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)] (L : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) {f : α -> E} {g : α -> F}, (MeasureTheory.AEStronglyMeasurable.{u1, u3} α E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) m f μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u4} α F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) m g μ) -> (MeasureTheory.AEStronglyMeasurable.{u1, u5} α G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) m (fun (x : α) => coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) => F -> G) (ContinuousLinearMap.toFun.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 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(ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F 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(NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) (fun (_x : ContinuousLinearMap.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) => E -> (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (ContinuousLinearMap.toFun.{u2, u2, u3, max u4 u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_5)) (ContinuousLinearMap.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u2, u2, u4, u5} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F G (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u4, u5} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u2, u2, u2, u4, u5} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10) (smulCommClass_self.{u2, u5} 𝕜 G (CommRing.toCommMonoid.{u2} 𝕜 (SeminormedCommRing.toCommRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (SeminormedAddCommGroup.toAddCommGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u2, u5} 𝕜 G (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u5} G (PseudoMetricSpace.toUniformSpace.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u2, u5} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (SeminormedAddGroup.toAddGroup.{u5} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))))) (SMulZeroClass.toHasSmul.{u2, u5} 𝕜 G (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (SMulWithZero.toSmulZeroClass.{u2, u5} 𝕜 G (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u5} 𝕜 G (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4)))))) (AddZeroClass.toHasZero.{u5} G (AddMonoid.toAddZeroClass.{u5} G (AddCommMonoid.toAddMonoid.{u5} G (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9))))) (Module.toMulActionWithZero.{u2, u5} 𝕜 G (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u5} G (NormedAddCommGroup.toAddCommGroup.{u5} G _inst_9)) (NormedSpace.toModule.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u2, u5} 𝕜 G (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9) _inst_10))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_4))))))) (LipschitzAdd.continuousAdd.{u5} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9)) (SubNegMonoid.toAddMonoid.{u5} G (AddGroup.toSubNegMonoid.{u5} G (NormedAddGroup.toAddGroup.{u5} G (NormedAddCommGroup.toNormedAddGroup.{u5} G _inst_9)))) (SeminormedAddCommGroup.to_lipschitzAdd.{u5} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G _inst_9))))) L (f x)) (g x)) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {𝕜 : Type.{u5}} [_inst_4 : NontriviallyNormedField.{u5} 𝕜] {E : Type.{u4}} [_inst_5 : NormedAddCommGroup.{u4} E] [_inst_6 : NormedSpace.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] {G : Type.{u3}} [_inst_9 : NormedAddCommGroup.{u3} G] [_inst_10 : NormedSpace.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)] (L : ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G 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(AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) 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(PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) m (fun (x : α) => FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F 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_inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G 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(NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) _x) (ContinuousMapClass.toFunLike.{max (max u4 u2) u3, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) E (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max (max u4 u2) u3, u5, u5, u4, max u2 u3} (ContinuousLinearMap.{u5, u5, u4, max u3 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u5, u4, max u2 u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_5)) (ContinuousLinearMap.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)) (ContinuousLinearMap.topologicalSpace.{u5, u5, u2, u3} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F G (SeminormedAddCommGroup.toAddCommGroup.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (SeminormedAddCommGroup.toAddCommGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))) (ContinuousLinearMap.addCommMonoid.{u5, u5, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedSpace.toModule.{u5, u4} 𝕜 E (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_5) _inst_6) (ContinuousLinearMap.module.{u5, u5, u5, u2, u3} 𝕜 𝕜 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u5, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10) (smulCommClass_self.{u5, u3} 𝕜 G (CommRing.toCommMonoid.{u5} 𝕜 (EuclideanDomain.toCommRing.{u5} 𝕜 (Field.toEuclideanDomain.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (MulActionWithZero.toMulAction.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10)))) (ContinuousSMul.continuousConstSMul.{u5, u3} 𝕜 G (UniformSpace.toTopologicalSpace.{u5} 𝕜 (PseudoMetricSpace.toUniformSpace.{u5} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (BoundedSMul.continuousSMul.{u5, u3} 𝕜 G (SeminormedRing.toPseudoMetricSpace.{u5} 𝕜 (SeminormedCommRing.toSeminormedRing.{u5} 𝕜 (NormedCommRing.toSeminormedCommRing.{u5} 𝕜 (NormedField.toNormedCommRing.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)) (CommMonoidWithZero.toZero.{u5} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u5} 𝕜 (Semifield.toCommGroupWithZero.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (NegZeroClass.toZero.{u3} G (SubNegZeroMonoid.toNegZeroClass.{u3} G (SubtractionMonoid.toSubNegZeroMonoid.{u3} G (SubtractionCommMonoid.toSubtractionMonoid.{u3} G (AddCommGroup.toDivisionAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))))) (SMulZeroClass.toSMul.{u5, u3} 𝕜 G (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (SMulWithZero.toSMulZeroClass.{u5, u3} 𝕜 G (MonoidWithZero.toZero.{u5} 𝕜 (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (MulActionWithZero.toSMulWithZero.{u5, u3} 𝕜 G (Semiring.toMonoidWithZero.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4)))))) (AddMonoid.toZero.{u3} G (AddCommMonoid.toAddMonoid.{u3} G (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)))) (Module.toMulActionWithZero.{u5, u3} 𝕜 G (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))) (AddCommGroup.toAddCommMonoid.{u3} G (NormedAddCommGroup.toAddCommGroup.{u3} G _inst_9)) (NormedSpace.toModule.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))))) (NormedSpace.boundedSMul.{u5, u3} 𝕜 G (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9) _inst_10))) (RingHom.id.{u5} 𝕜 (Semiring.toNonAssocSemiring.{u5} 𝕜 (DivisionSemiring.toSemiring.{u5} 𝕜 (Semifield.toDivisionSemiring.{u5} 𝕜 (Field.toSemifield.{u5} 𝕜 (NormedField.toField.{u5} 𝕜 (NontriviallyNormedField.toNormedField.{u5} 𝕜 _inst_4))))))) (TopologicalAddGroup.toContinuousAdd.{u3} G (UniformSpace.toTopologicalSpace.{u3} G (PseudoMetricSpace.toUniformSpace.{u3} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9)))) (NormedAddGroup.toAddGroup.{u3} G (NormedAddCommGroup.toNormedAddGroup.{u3} G _inst_9)) (SeminormedAddCommGroup.to_topologicalAddGroup.{u3} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} G _inst_9))))))) L (f x)) (g x)) μ)
+Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂ₓ'. -/
+theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
+ {g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
+ AEStronglyMeasurable (fun x => L (f x) (g x)) μ :=
+ L.continuous₂.comp_aestronglyMeasurable <| hf.prod_mk hg
+#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂
end ContinuousLinearMapNontriviallyNormedField
-theorem aeStronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+/- warning: ae_strongly_measurable_with_density_iff -> aestronglyMeasurable_withDensity_iff is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {E : Type.{u2}} [_inst_4 : NormedAddCommGroup.{u2} E] [_inst_5 : NormedSpace.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)] {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> E}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x)))) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m (fun (x : α) => SMul.smul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))))) (Module.toMulActionWithZero.{0, u2} Real E (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4))) (NormedSpace.toModule.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4) _inst_5))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (f x)) (g x)) μ))
+but is expected to have type
+ forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {E : Type.{u2}} [_inst_4 : NormedAddCommGroup.{u2} E] [_inst_5 : NormedSpace.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)] {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> E}, Iff (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => ENNReal.some (f x)))) (MeasureTheory.AEStronglyMeasurable.{u1, u2} α E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4)))) m (fun (x : α) => HSMul.hSMul.{0, u2, u2} Real E E (instHSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_4)) (NormedSpace.toModule.{0, u2} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_4) _inst_5)))))) (NNReal.toReal (f x)) (g x)) μ))
+Case conversion may be inaccurate. Consider using '#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iffₓ'. -/
+theorem aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
- AeStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
- AeStronglyMeasurable (fun x => (f x : ℝ) • g x) μ :=
+ AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
+ AEStronglyMeasurable (fun x => (f x : ℝ) • g x) μ :=
by
constructor
· rintro ⟨g', g'meas, hg'⟩
@@ -1878,7 +2714,7 @@ theorem aeStronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
-#align ae_strongly_measurable_with_density_iff aeStronglyMeasurable_withDensity_iff
+#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iff
end AeStronglyMeasurable
@@ -1893,59 +2729,105 @@ section Mk
variable [Zero β]
+#print MeasureTheory.AEFinStronglyMeasurable.mk /-
/-- A `fin_strongly_measurable` function such that `f =ᵐ[μ] hf.mk f`. See lemmas
`fin_strongly_measurable_mk` and `ae_eq_mk`. -/
-protected noncomputable def mk (f : α → β) (hf : AeFinStronglyMeasurable f μ) : α → β :=
+protected noncomputable def mk (f : α → β) (hf : AEFinStronglyMeasurable f μ) : α → β :=
hf.some
-#align measure_theory.ae_fin_strongly_measurable.mk MeasureTheory.AeFinStronglyMeasurable.mk
+#align measure_theory.ae_fin_strongly_measurable.mk MeasureTheory.AEFinStronglyMeasurable.mk
+-/
-theorem finStronglyMeasurable_mk (hf : AeFinStronglyMeasurable f μ) :
+/- warning: measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk -> MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), MeasureTheory.FinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (MeasureTheory.AEFinStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 _inst_3 f hf) μ
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), MeasureTheory.FinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m (MeasureTheory.AEFinStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 _inst_3 f hf) μ
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mkₓ'. -/
+theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
FinStronglyMeasurable (hf.mk f) μ :=
hf.choose_spec.1
-#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AeFinStronglyMeasurable.finStronglyMeasurable_mk
-
-theorem ae_eq_mk (hf : AeFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
+#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
+
+/- warning: measure_theory.ae_fin_strongly_measurable.ae_eq_mk -> MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f (MeasureTheory.AEFinStronglyMeasurable.mk.{u1, u2} α β m μ _inst_2 _inst_3 f hf)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f (MeasureTheory.AEFinStronglyMeasurable.mk.{u2, u1} α β m μ _inst_2 _inst_3 f hf)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mkₓ'. -/
+theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AeFinStronglyMeasurable.ae_eq_mk
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
-protected theorem aEMeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
- [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AeFinStronglyMeasurable f μ) :
+#print MeasureTheory.AEFinStronglyMeasurable.aemeasurable /-
+protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
+ [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
AEMeasurable f μ :=
⟨hf.mk f, hf.finStronglyMeasurable_mk.Measurable, hf.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.ae_measurable MeasureTheory.AeFinStronglyMeasurable.aEMeasurable
+#align measure_theory.ae_fin_strongly_measurable.ae_measurable MeasureTheory.AEFinStronglyMeasurable.aemeasurable
+-/
end Mk
section Arithmetic
-protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AeFinStronglyMeasurable f μ)
- (hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f * g) μ :=
+/- warning: measure_theory.ae_fin_strongly_measurable.mul -> MeasureTheory.AEFinStronglyMeasurable.mul is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : MonoidWithZero.{u2} β] [_inst_4 : ContinuousMul.{u2} β _inst_2 (MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3)))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (MonoidWithZero.toZero.{u2} β _inst_3) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHMul.{max u1 u2} (α -> β) (Pi.instMul.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mulₓ'. -/
+protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
+ (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AeFinStronglyMeasurable.mul
-
-protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AeFinStronglyMeasurable f μ)
- (hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f + g) μ :=
+#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mul
+
+/- warning: measure_theory.ae_fin_strongly_measurable.add -> MeasureTheory.AEFinStronglyMeasurable.add is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddMonoid.{u2} β] [_inst_4 : ContinuousAdd.{u2} β _inst_2 (AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_3) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHAdd.{max u1 u2} (α -> β) (Pi.instAdd.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.addₓ'. -/
+protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
+ (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ :=
⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.add hg.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AeFinStronglyMeasurable.add
-
-protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AeFinStronglyMeasurable f μ) :
- AeFinStronglyMeasurable (-f) μ :=
+#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.add
+
+/- warning: measure_theory.ae_fin_strongly_measurable.neg -> MeasureTheory.AEFinStronglyMeasurable.neg is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasNeg.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))) f) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : TopologicalAddGroup.{u2} β _inst_2 _inst_3], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m (Neg.neg.{max u1 u2} (α -> β) (Pi.instNeg.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => NegZeroClass.toNeg.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3))))) f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.negₓ'. -/
+protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
+ AEFinStronglyMeasurable (-f) μ :=
⟨-hf.mk f, hf.finStronglyMeasurable_mk.neg, hf.ae_eq_mk.neg⟩
-#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AeFinStronglyMeasurable.neg
-
-protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AeFinStronglyMeasurable f μ)
- (hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f - g) μ :=
+#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.neg
+
+/- warning: measure_theory.ae_fin_strongly_measurable.sub -> MeasureTheory.AEFinStronglyMeasurable.sub is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (SubNegMonoid.toAddMonoid.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) m (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toHasSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : AddGroup.{u2} β] [_inst_4 : ContinuousSub.{u2} β _inst_2 (SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3))], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (NegZeroClass.toZero.{u2} β (SubNegZeroMonoid.toNegZeroClass.{u2} β (SubtractionMonoid.toSubNegZeroMonoid.{u2} β (AddGroup.toSubtractionMonoid.{u2} β _inst_3)))) m (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (α -> β) (α -> β) (α -> β) (instHSub.{max u1 u2} (α -> β) (Pi.instSub.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SubNegMonoid.toSub.{u2} β (AddGroup.toSubNegMonoid.{u2} β _inst_3)))) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.subₓ'. -/
+protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
+ (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ :=
⟨hf.mk f - hg.mk g, hf.finStronglyMeasurable_mk.sub hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AeFinStronglyMeasurable.sub
-
+#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.sub
+
+/- warning: measure_theory.ae_fin_strongly_measurable.const_smul -> MeasureTheory.AEFinStronglyMeasurable.const_smul is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m f μ) -> (forall (c : 𝕜), MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) m (SMul.smul.{u3, max u1 u2} 𝕜 (α -> β) (Function.hasSMul.{u1, u3, u2} α 𝕜 β (SMulZeroClass.toHasSmul.{u3, u2} 𝕜 β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_4)) (DistribSMul.toSmulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6)))) c f) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {𝕜 : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} 𝕜] [_inst_4 : AddMonoid.{u2} β] [_inst_5 : Monoid.{u3} 𝕜] [_inst_6 : DistribMulAction.{u3, u2} 𝕜 β _inst_5 _inst_4] [_inst_7 : ContinuousSMul.{u3, u2} 𝕜 β (SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))) _inst_3 _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m f μ) -> (forall (c : 𝕜), MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 (AddMonoid.toZero.{u2} β _inst_4) m (HSMul.hSMul.{u3, max u1 u2, max u1 u2} 𝕜 (α -> β) (α -> β) (instHSMul.{u3, max u1 u2} 𝕜 (α -> β) (Pi.instSMul.{u1, u2, u3} α 𝕜 (fun (a._@.Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic._hyg.21457 : α) => β) (fun (i : α) => SMulZeroClass.toSMul.{u3, u2} 𝕜 β (AddMonoid.toZero.{u2} β _inst_4) (DistribSMul.toSMulZeroClass.{u3, u2} 𝕜 β (AddMonoid.toAddZeroClass.{u2} β _inst_4) (DistribMulAction.toDistribSMul.{u3, u2} 𝕜 β _inst_5 _inst_4 _inst_6))))) c f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AEFinStronglyMeasurable.const_smulₓ'. -/
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
- [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AeFinStronglyMeasurable f μ) (c : 𝕜) :
- AeFinStronglyMeasurable (c • f) μ :=
+ [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
+ AEFinStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.finStronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
-#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AeFinStronglyMeasurable.const_smul
+#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AEFinStronglyMeasurable.const_smul
end Arithmetic
@@ -1953,23 +2835,41 @@ section Order
variable [Zero β]
-protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AeFinStronglyMeasurable f μ)
- (hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f ⊔ g) μ :=
+/- warning: measure_theory.ae_fin_strongly_measurable.sup -> MeasureTheory.AEFinStronglyMeasurable.sup is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toHasSup.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Sup.sup.{max u1 u2} (α -> β) (Pi.hasSup.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toHasSup.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeSup.{u2} β] [_inst_5 : ContinuousSup.{u2} β _inst_2 (SemilatticeSup.toSup.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Sup.sup.{max u2 u1} (α -> β) (Pi.instSupForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeSup.toSup.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.supₓ'. -/
+protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
+ (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AeFinStronglyMeasurable.sup
-
-protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AeFinStronglyMeasurable f μ)
- (hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f ⊓ g) μ :=
+#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.sup
+
+/- warning: measure_theory.ae_fin_strongly_measurable.inf -> MeasureTheory.AEFinStronglyMeasurable.inf is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toHasInf.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Inf.inf.{max u1 u2} (α -> β) (Pi.hasInf.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toHasInf.{u2} β _inst_4)) f g) μ)
+but is expected to have type
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {g : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : SemilatticeInf.{u2} β] [_inst_5 : ContinuousInf.{u2} β _inst_2 (SemilatticeInf.toInf.{u2} β _inst_4)], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m g μ) -> (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m (Inf.inf.{max u2 u1} (α -> β) (Pi.instInfForAll.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => SemilatticeInf.toInf.{u2} β _inst_4)) f g) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AEFinStronglyMeasurable.infₓ'. -/
+protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
+ (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.inf hg.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AeFinStronglyMeasurable.inf
+#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AEFinStronglyMeasurable.inf
end Order
variable [Zero β] [T2Space β]
-theorem exists_set_sigmaFinite (hf : AeFinStronglyMeasurable f μ) :
+/- warning: measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite -> MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (MeasurableSet.{u1} α m t) (And (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t))) f (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))) (MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t)))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2], (MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ) -> (Exists.{succ u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (MeasurableSet.{u2} α m t) (And (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) t))) f (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => β) (fun (i : α) => _inst_3))))) (MeasureTheory.SigmaFinite.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ t)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFiniteₓ'. -/
+theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict (tᶜ)] 0 ∧ SigmaFinite (μ.restrict t) :=
by
rcases hf with ⟨g, hg, hfg⟩
@@ -1978,27 +2878,43 @@ theorem exists_set_sigmaFinite (hf : AeFinStronglyMeasurable f μ) :
refine' eventually_eq.trans (ae_restrict_of_ae hfg) _
rw [eventually_eq, ae_restrict_iff' ht.compl]
exact eventually_of_forall hgt_zero
-#align measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.AeFinStronglyMeasurable.exists_set_sigmaFinite
+#align measure_theory.ae_fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite
+#print MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet /-
/-- A measurable set `t` such that `f =ᵐ[μ.restrict tᶜ] 0` and `sigma_finite (μ.restrict t)`. -/
-def sigmaFiniteSet (hf : AeFinStronglyMeasurable f μ) : Set α :=
+def sigmaFiniteSet (hf : AEFinStronglyMeasurable f μ) : Set α :=
hf.exists_set_sigmaFinite.some
-#align measure_theory.ae_fin_strongly_measurable.sigma_finite_set MeasureTheory.AeFinStronglyMeasurable.sigmaFiniteSet
+#align measure_theory.ae_fin_strongly_measurable.sigma_finite_set MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet
+-/
-protected theorem measurableSet (hf : AeFinStronglyMeasurable f μ) :
+/- warning: measure_theory.ae_fin_strongly_measurable.measurable_set -> MeasureTheory.AEFinStronglyMeasurable.measurableSet is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), MeasurableSet.{u1} α m (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u1, u2} α β m μ _inst_2 f _inst_3 _inst_4 hf)
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), MeasurableSet.{u2} α m (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u2, u1} α β m μ _inst_2 f _inst_3 _inst_4 hf)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSetₓ'. -/
+protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
MeasurableSet hf.sigmaFiniteSet :=
hf.exists_set_sigmaFinite.choose_spec.1
-#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AeFinStronglyMeasurable.measurableSet
-
-theorem ae_eq_zero_compl (hf : AeFinStronglyMeasurable f μ) :
+#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSet
+
+/- warning: measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl -> MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} [_inst_3 : Zero.{u2} β] [_inst_4 : T2Space.{u2} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u1, u2} α β m μ _inst_2 f _inst_3 _inst_4 hf)))) f (OfNat.ofNat.{max u1 u2} (α -> β) 0 (OfNat.mk.{max u1 u2} (α -> β) 0 (Zero.zero.{max u1 u2} (α -> β) (Pi.instZero.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_3)))))
+but is expected to have type
+ forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} [_inst_3 : Zero.{u1} β] [_inst_4 : T2Space.{u1} β _inst_2] (hf : MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α β _inst_2 _inst_3 m f μ), Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m (MeasureTheory.Measure.restrict.{u2} α m μ (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) (MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet.{u2, u1} α β m μ _inst_2 f _inst_3 _inst_4 hf)))) f (OfNat.ofNat.{max u2 u1} (α -> β) 0 (Zero.toOfNat0.{max u2 u1} (α -> β) (Pi.instZero.{u2, u1} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => β) (fun (i : α) => _inst_3))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_complₓ'. -/
+theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
f =ᵐ[μ.restrict (hf.sigmaFiniteSetᶜ)] 0 :=
hf.exists_set_sigmaFinite.choose_spec.2.1
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AeFinStronglyMeasurable.ae_eq_zero_compl
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
-instance sigmaFinite_restrict (hf : AeFinStronglyMeasurable f μ) :
+#print MeasureTheory.AEFinStronglyMeasurable.sigmaFinite_restrict /-
+instance sigmaFinite_restrict (hf : AEFinStronglyMeasurable f μ) :
SigmaFinite (μ.restrict hf.sigmaFiniteSet) :=
hf.exists_set_sigmaFinite.choose_spec.2.2
-#align measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict MeasureTheory.AeFinStronglyMeasurable.sigmaFinite_restrict
+#align measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict MeasureTheory.AEFinStronglyMeasurable.sigmaFinite_restrict
+-/
end AeFinStronglyMeasurable
@@ -2008,6 +2924,12 @@ variable {G : Type _} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : Measure
[SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G]
{f : α → G}
+/- warning: measure_theory.fin_strongly_measurable_iff_measurable -> MeasureTheory.finStronglyMeasurable_iff_measurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {G : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} G] [_inst_3 : MeasurableSpace.{u2} G] [_inst_4 : BorelSpace.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u1} α m0 μ], Iff (MeasureTheory.FinStronglyMeasurable.{u1, u2} α G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G (SeminormedAddGroup.toAddGroup.{u2} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} G _inst_2)))))) m0 f μ) (Measurable.{u1, u2} α G m0 _inst_3 f)
+but is expected to have type
+ forall {α : Type.{u2}} {G : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} G] [_inst_3 : MeasurableSpace.{u1} G] [_inst_4 : BorelSpace.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u2} α m0 μ], Iff (MeasureTheory.FinStronglyMeasurable.{u2, u1} α G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (SubtractionCommMonoid.toSubtractionMonoid.{u1} G (AddCommGroup.toDivisionAddCommMonoid.{u1} G (SeminormedAddCommGroup.toAddCommGroup.{u1} G _inst_2)))))) m0 f μ) (Measurable.{u2, u1} α G m0 _inst_3 f)
+Case conversion may be inaccurate. Consider using '#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurableₓ'. -/
/-- In a space with second countable topology and a sigma-finite measure, `fin_strongly_measurable`
and `measurable` are equivalent. -/
theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Measure α)
@@ -2015,15 +2937,27 @@ theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Mea
⟨fun h => h.Measurable, fun h => (Measurable.stronglyMeasurable h).FinStronglyMeasurable μ⟩
#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurable
+/- warning: measure_theory.ae_fin_strongly_measurable_iff_ae_measurable -> MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {G : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} G] [_inst_3 : MeasurableSpace.{u2} G] [_inst_4 : BorelSpace.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u1} α m0 μ], Iff (MeasureTheory.AEFinStronglyMeasurable.{u1, u2} α G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G _inst_2))) (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G (SeminormedAddGroup.toAddGroup.{u2} G (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} G _inst_2)))))) m0 f μ) (AEMeasurable.{u1, u2} α G _inst_3 m0 f μ)
+but is expected to have type
+ forall {α : Type.{u2}} {G : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} G] [_inst_3 : MeasurableSpace.{u1} G] [_inst_4 : BorelSpace.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) _inst_3] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2)))] {f : α -> G} {m0 : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m0) [_inst_6 : MeasureTheory.SigmaFinite.{u2} α m0 μ], Iff (MeasureTheory.AEFinStronglyMeasurable.{u2, u1} α G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G _inst_2))) (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (SubtractionCommMonoid.toSubtractionMonoid.{u1} G (AddCommGroup.toDivisionAddCommMonoid.{u1} G (SeminormedAddCommGroup.toAddCommGroup.{u1} G _inst_2)))))) m0 f μ) (AEMeasurable.{u2, u1} α G _inst_3 m0 f μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aefinStronglyMeasurable_iff_aemeasurableₓ'. -/
/-- In a space with second countable topology and a sigma-finite measure,
`ae_fin_strongly_measurable` and `ae_measurable` are equivalent. -/
-theorem aeFinStronglyMeasurable_iff_aEMeasurable {m0 : MeasurableSpace α} (μ : Measure α)
- [SigmaFinite μ] : AeFinStronglyMeasurable f μ ↔ AEMeasurable f μ := by
+theorem aefinStronglyMeasurable_iff_aemeasurable {m0 : MeasurableSpace α} (μ : Measure α)
+ [SigmaFinite μ] : AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ := by
simp_rw [ae_fin_strongly_measurable, AEMeasurable, fin_strongly_measurable_iff_measurable]
-#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aeFinStronglyMeasurable_iff_aEMeasurable
+#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable
end SecondCountableTopology
+/- warning: measure_theory.measurable_uncurry_of_continuous_of_measurable -> MeasureTheory.measurable_uncurry_of_continuous_of_measurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u3} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u3} ι _inst_2] [_inst_4 : MeasurableSpace.{u3} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u3} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u3} ι _inst_2 _inst_4] {mβ : MeasurableSpace.{u2} β} [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : BorelSpace.{u2} β _inst_7 mβ] {m : MeasurableSpace.{u1} α} {u : ι -> α -> β}, (forall (x : α), Continuous.{u3, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), Measurable.{u1, u2} α β m mβ (u i)) -> (Measurable.{max u3 u1, u2} (Prod.{u3, u1} ι α) β (Prod.instMeasurableSpace.{u3, u1} ι α _inst_4 m) mβ (Function.uncurry.{u3, u1, u2} ι α β u))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u1} ι _inst_2] [_inst_4 : MeasurableSpace.{u1} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u1} ι _inst_2 _inst_4] {mβ : MeasurableSpace.{u2} β} [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : BorelSpace.{u2} β _inst_7 mβ] {m : MeasurableSpace.{u3} α} {u : ι -> α -> β}, (forall (x : α), Continuous.{u1, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), Measurable.{u3, u2} α β m mβ (u i)) -> (Measurable.{max u3 u1, u2} (Prod.{u1, u3} ι α) β (Prod.instMeasurableSpace.{u1, u3} ι α _inst_4 m) mβ (Function.uncurry.{u1, u3, u2} ι α β u))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurableₓ'. -/
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [TopologicalSpace ι]
[MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
{mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
@@ -2060,6 +2994,12 @@ theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [Topo
exact ((t_sf n).Measurable.comp measurable_fst).subtype_mk
#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurable
+/- warning: measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable -> MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} {β : Type.{u2}} {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u3} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u3} ι _inst_2] [_inst_4 : MeasurableSpace.{u3} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u3} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u3} ι _inst_2 _inst_4] [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : MeasurableSpace.{u1} α] {u : ι -> α -> β}, (forall (x : α), Continuous.{u3, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), MeasureTheory.StronglyMeasurable.{u1, u2} α β _inst_7 _inst_9 (u i)) -> (MeasureTheory.StronglyMeasurable.{max u3 u1, u2} (Prod.{u3, u1} ι α) β _inst_7 (Prod.instMeasurableSpace.{u3, u1} ι α _inst_4 _inst_9) (Function.uncurry.{u3, u1, u2} ι α β u))
+but is expected to have type
+ forall {α : Type.{u3}} {β : Type.{u2}} {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} ι] [_inst_3 : TopologicalSpace.MetrizableSpace.{u1} ι _inst_2] [_inst_4 : MeasurableSpace.{u1} ι] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} ι _inst_2] [_inst_6 : OpensMeasurableSpace.{u1} ι _inst_2 _inst_4] [_inst_7 : TopologicalSpace.{u2} β] [_inst_8 : TopologicalSpace.PseudoMetrizableSpace.{u2} β _inst_7] [_inst_9 : MeasurableSpace.{u3} α] {u : ι -> α -> β}, (forall (x : α), Continuous.{u1, u2} ι β _inst_2 _inst_7 (fun (i : ι) => u i x)) -> (forall (i : ι), MeasureTheory.StronglyMeasurable.{u3, u2} α β _inst_7 _inst_9 (u i)) -> (MeasureTheory.StronglyMeasurable.{max u3 u1, u2} (Prod.{u1, u3} ι α) β _inst_7 (Prod.instMeasurableSpace.{u1, u3} ι α _inst_4 _inst_9) (Function.uncurry.{u1, u3, u2} ι α β u))
+Case conversion may be inaccurate. Consider using '#align measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurableₓ'. -/
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type _}
[TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
[OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -612,7 +612,7 @@ end CommMonoid
theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : StronglyMeasurable f) :
TopologicalSpace.IsSeparable (range f) :=
by
- have : is_separable (closure (⋃ n, range (hf.approx n))) :=
+ have : IsSeparable (closure (⋃ n, range (hf.approx n))) :=
(is_separable_Union fun n => (simple_func.finite_range (hf.approx n)).IsSeparable).closure
apply this.mono
rintro _ ⟨x, rfl⟩
@@ -728,7 +728,7 @@ theorem Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [Topologi
exact mem_range_self x }
have : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
exact hG.measurable_embedding.measurable_comp_iff.1 this
- · have : is_separable (g ⁻¹' range (g ∘ f)) := hg.is_separable_preimage H.is_separable_range
+ · have : IsSeparable (g ⁻¹' range (g ∘ f)) := hg.is_separable_preimage H.is_separable_range
convert this
ext x
simp [hg.inj.eq_iff]
@@ -743,7 +743,7 @@ theorem stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α} [To
refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩
· exact measurable_of_tendsto_metrizable' u (fun i => (hf i).Measurable) limUnder
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
- have : is_separable (closure (⋃ i, range (f (v i)))) :=
+ have : IsSeparable (closure (⋃ i, range (f (v i)))) :=
(is_separable_Union fun i => (hf (v i)).isSeparable_range).closure
apply this.mono
rintro _ ⟨x, rfl⟩
@@ -1691,7 +1691,7 @@ theorem aeStronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace
refine' aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩
· exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).AEMeasurable) limUnder
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
- have : ∀ n : ℕ, ∃ t : Set β, is_separable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
+ have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
(aeStronglyMeasurable_iff_aEMeasurable_separable.1 (hf (v n))).2
choose t t_sep ht using this
refine' ⟨closure (⋃ i, t i), (is_separable_Union fun i => t_sep i).closure, _⟩
@@ -1727,7 +1727,7 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
refine'
aeStronglyMeasurable_iff_aEMeasurable_separable.2
⟨AEMeasurable.sum_measure fun i => (h i).AEMeasurable, _⟩
- have A : ∀ i : ι, ∃ t : Set β, is_separable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
+ have A : ∀ i : ι, ∃ t : Set β, IsSeparable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
(aeStronglyMeasurable_iff_aEMeasurable_separable.1 (h i)).2
choose t t_sep ht using A
refine' ⟨⋃ i, t i, is_separable_Union t_sep, _⟩
@@ -2088,7 +2088,7 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
rfl
rw [this, measurable_swap_iff]
exact measurable_from_prod_countable fun j => (h j).Measurable
- · have : is_separable (⋃ i : (t_sf n).range, range (u i)) :=
+ · have : IsSeparable (⋃ i : (t_sf n).range, range (u i)) :=
is_separable_Union fun i => (h i).isSeparable_range
apply this.mono
rintro _ ⟨⟨i, x⟩, rfl⟩
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -1689,7 +1689,7 @@ theorem aeStronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace
AeStronglyMeasurable g μ := by
borelize β
refine' aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩
- · exact aEMeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).AEMeasurable) limUnder
+ · exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).AEMeasurable) limUnder
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, is_separable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
(aeStronglyMeasurable_iff_aEMeasurable_separable.1 (hf (v n))).2
mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit 7317149f12f55affbc900fc873d0d422485122b9
+! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
+import Mathbin.Analysis.NormedSpace.FiniteDimension
import Mathbin.Analysis.NormedSpace.BoundedLinearMaps
-import Mathbin.Topology.MetricSpace.Metrizable
+import Mathbin.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathbin.MeasureTheory.Integral.Lebesgue
import Mathbin.MeasureTheory.Function.SimpleFuncDense
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -341,7 +341,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
· exact ⟨n, fun m hnm => Set.mem_inter (hn m hnm) hxt⟩
rsuffices ⟨n, hn⟩ : ∃ n, x ∈ S n
· exact ⟨n, fun m hnm => monotone_spanning_sets (μ.restrict t) hnm hn⟩
- rw [← Set.mem_unionᵢ, Union_spanning_sets (μ.restrict t)]
+ rw [← Set.mem_iUnion, Union_spanning_sets (μ.restrict t)]
trivial
refine' ⟨n, fun m hnm => _⟩
simp_rw [fs, simple_func.restrict_apply _ ((hS_meas m).inter ht),
@@ -993,7 +993,7 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
have norm_sets_spanning : (⋃ n, norm_sets n) = Set.univ :=
by
ext1 x
- simp only [Set.mem_unionᵢ, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
+ simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
exact ⟨⌈‖f x‖⌉₊, Nat.le_ceil ‖f x‖⟩
let sets n := sigma_finite_sets n ∩ norm_sets n
have h_meas : ∀ n, measurable_set[m] (sets n) :=
@@ -1010,7 +1010,7 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
· have :
(⋃ i, sigma_finite_sets i ∩ norm_sets i) = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i :=
by
- refine' Set.unionᵢ_inter_of_monotone (monotone_spanning_sets (μ.trim hm)) fun i j hij x => _
+ refine' Set.iUnion_inter_of_monotone (monotone_spanning_sets (μ.trim hm)) fun i j hij x => _
simp only [norm_sets, Set.mem_setOf_eq]
refine' fun hif => hif.trans _
exact_mod_cast hij
@@ -1073,20 +1073,20 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
let T n := support (fs n)
have hT_meas : ∀ n, MeasurableSet (T n) := fun n => simple_func.measurable_set_support (fs n)
let t := ⋃ n, T n
- refine' ⟨t, MeasurableSet.unionᵢ hT_meas, _, _⟩
+ refine' ⟨t, MeasurableSet.iUnion hT_meas, _, _⟩
· have h_fs_zero : ∀ n, ∀ x ∈ tᶜ, fs n x = 0 :=
by
intro n x hxt
- rw [Set.mem_compl_iff, Set.mem_unionᵢ, not_exists] at hxt
+ rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
simpa using hxt n
refine' fun x hxt => tendsto_nhds_unique (h_approx x) _
rw [funext fun n => h_fs_zero n x hxt]
exact tendsto_const_nhds
· refine' ⟨⟨⟨fun n => tᶜ ∪ T n, fun n => trivial, fun n => _, _⟩⟩⟩
- · rw [measure.restrict_apply' (MeasurableSet.unionᵢ hT_meas), Set.union_inter_distrib_right,
+ · rw [measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,
Set.compl_inter_self t, Set.empty_union]
exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n)
- · rw [← Set.union_unionᵢ (tᶜ) T]
+ · rw [← Set.union_iUnion (tᶜ) T]
exact Set.compl_union_self _
#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
@@ -1756,25 +1756,25 @@ theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
aeStronglyMeasurable_add_measure_iff.2 ⟨hμ, hν⟩
#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AeStronglyMeasurable.add_measure
-protected theorem unionᵢ [PseudoMetrizableSpace β] {s : ι → Set α}
+protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
(h : ∀ i, AeStronglyMeasurable f (μ.restrict (s i))) :
AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
- (sum_measure h).mono_measure <| restrict_unionᵢ_le
-#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AeStronglyMeasurable.unionᵢ
+ (sum_measure h).mono_measure <| restrict_iUnion_le
+#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AeStronglyMeasurable.iUnion
@[simp]
-theorem aeStronglyMeasurable_unionᵢ_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
+theorem aeStronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
∀ i, AeStronglyMeasurable f (μ.restrict (s i)) :=
- ⟨fun h i => h.mono_measure <| restrict_mono (subset_unionᵢ _ _) le_rfl,
- AeStronglyMeasurable.unionᵢ⟩
-#align ae_strongly_measurable_Union_iff aeStronglyMeasurable_unionᵢ_iff
+ ⟨fun h i => h.mono_measure <| restrict_mono (subset_iUnion _ _) le_rfl,
+ AeStronglyMeasurable.iUnion⟩
+#align ae_strongly_measurable_Union_iff aeStronglyMeasurable_iUnion_iff
@[simp]
theorem aeStronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α} :
AeStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
AeStronglyMeasurable f (μ.restrict s) ∧ AeStronglyMeasurable f (μ.restrict t) :=
- by simp only [union_eq_Union, aeStronglyMeasurable_unionᵢ_iff, Bool.forall_bool, cond, and_comm]
+ by simp only [union_eq_Union, aeStronglyMeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]
#align ae_strongly_measurable_union_iff aeStronglyMeasurable_union_iff
theorem aeStronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa
@@ -301,9 +301,10 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
-theorem finStronglyMeasurableOfSetSigmaFinite [TopologicalSpace β] [Zero β] {m : MeasurableSpace α}
- {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α} (ht : MeasurableSet t)
- (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) : FinStronglyMeasurable f μ :=
+theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
+ {m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
+ (ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
+ FinStronglyMeasurable f μ :=
by
haveI : sigma_finite (μ.restrict t) := htμ
let S := spanning_sets (μ.restrict t)
@@ -351,13 +352,13 @@ theorem finStronglyMeasurableOfSetSigmaFinite [TopologicalSpace β] [Zero β] {m
refine' ⟨max n₁ n₂, fun m hm => _⟩
rw [hn₁ m ((le_max_left _ _).trans hm.le)]
exact hn₂ m ((le_max_right _ _).trans hm.le)
-#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurableOfSetSigmaFinite
+#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
/-- If the measure is sigma-finite, all strongly measurable functions are
`fin_strongly_measurable`. -/
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
(hf : StronglyMeasurable f) (μ : Measure α) [SigmaFinite μ] : FinStronglyMeasurable f μ :=
- hf.finStronglyMeasurableOfSetSigmaFinite MeasurableSet.univ (by simp)
+ hf.finStronglyMeasurable_of_set_sigmaFinite MeasurableSet.univ (by simp)
(by rwa [measure.restrict_univ])
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
@@ -369,11 +370,11 @@ protected theorem measurable {m : MeasurableSpace α} [TopologicalSpace β] [Pse
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
/-- A strongly measurable function is almost everywhere measurable. -/
-protected theorem aeMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
+protected theorem aEMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
- (hf : StronglyMeasurable f) : AeMeasurable f μ :=
- hf.Measurable.AeMeasurable
-#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aeMeasurable
+ (hf : StronglyMeasurable f) : AEMeasurable f μ :=
+ hf.Measurable.AEMeasurable
+#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aEMeasurable
theorem Continuous.comp_stronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
[TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
@@ -417,13 +418,13 @@ theorem comp_measurable [TopologicalSpace β] {m : MeasurableSpace α} {m' : Mea
theorem of_uncurry_left [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) :=
- hf.compMeasurable measurable_prod_mk_left
+ hf.comp_measurable measurable_prod_mk_left
#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_left
theorem of_uncurry_right [TopologicalSpace β] {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
{f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
StronglyMeasurable fun x => f x y :=
- hf.compMeasurable measurable_prod_mk_right
+ hf.comp_measurable measurable_prod_mk_right
#align measure_theory.strongly_measurable.of_uncurry_right MeasureTheory.StronglyMeasurable.of_uncurry_right
section Arithmetic
@@ -1021,13 +1022,13 @@ end StronglyMeasurable
/-! ## Finitely strongly measurable functions -/
-theorem finStronglyMeasurableZero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
+theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
[TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ :=
⟨0, by
simp only [Pi.zero_apply, simple_func.coe_zero, support_zero', measure_empty,
WithTop.zero_lt_top, forall_const],
fun n => tendsto_const_nhds⟩
-#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurableZero
+#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zero
namespace FinStronglyMeasurable
@@ -1136,14 +1137,14 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
-protected theorem constSmul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜] [DistribMulAction 𝕜 β]
- [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
+protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
+ [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
FinStronglyMeasurable (c • f) μ :=
by
refine' ⟨fun n => c • hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c⟩
rw [simple_func.coe_smul]
refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n)
-#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.constSmul
+#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smul
end Arithmetic
@@ -1181,29 +1182,29 @@ theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
StronglyMeasurable f ∧
∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) :=
⟨fun hf => ⟨hf.StronglyMeasurable, hf.exists_set_sigmaFinite⟩, fun hf =>
- hf.1.finStronglyMeasurableOfSetSigmaFinite hf.2.choose_spec.1 hf.2.choose_spec.2.1
+ hf.1.finStronglyMeasurable_of_set_sigmaFinite hf.2.choose_spec.1 hf.2.choose_spec.2.1
hf.2.choose_spec.2.2⟩
#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
-theorem aeFinStronglyMeasurableZero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
+theorem aeFinStronglyMeasurable_zero {α β} {m : MeasurableSpace α} (μ : Measure α) [Zero β]
[TopologicalSpace β] : AeFinStronglyMeasurable (0 : α → β) μ :=
- ⟨0, finStronglyMeasurableZero, EventuallyEq.rfl⟩
-#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aeFinStronglyMeasurableZero
+ ⟨0, finStronglyMeasurable_zero, EventuallyEq.rfl⟩
+#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aeFinStronglyMeasurable_zero
/-! ## Almost everywhere strongly measurable functions -/
-theorem aeStronglyMeasurableConst {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
- {b : β} : AeStronglyMeasurable (fun a : α => b) μ :=
+theorem aeStronglyMeasurable_const {α β} {m : MeasurableSpace α} {μ : Measure α}
+ [TopologicalSpace β] {b : β} : AeStronglyMeasurable (fun a : α => b) μ :=
stronglyMeasurable_const.AeStronglyMeasurable
-#align measure_theory.ae_strongly_measurable_const MeasureTheory.aeStronglyMeasurableConst
+#align measure_theory.ae_strongly_measurable_const MeasureTheory.aeStronglyMeasurable_const
@[to_additive]
-theorem aeStronglyMeasurableOne {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
+theorem aeStronglyMeasurable_one {α β} {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
[One β] : AeStronglyMeasurable (1 : α → β) μ :=
stronglyMeasurable_one.AeStronglyMeasurable
-#align measure_theory.ae_strongly_measurable_one MeasureTheory.aeStronglyMeasurableOne
-#align measure_theory.ae_strongly_measurable_zero MeasureTheory.ae_strongly_measurable_zero
+#align measure_theory.ae_strongly_measurable_one MeasureTheory.aeStronglyMeasurable_one
+#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aeStronglyMeasurable_zero
@[simp]
theorem Subsingleton.aeStronglyMeasurable {m : MeasurableSpace α} [TopologicalSpace β]
@@ -1212,19 +1213,19 @@ theorem Subsingleton.aeStronglyMeasurable {m : MeasurableSpace α} [TopologicalS
#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aeStronglyMeasurable
@[simp]
-theorem Subsingleton.aeStronglyMeasurable' {m : MeasurableSpace α} [TopologicalSpace β]
+theorem Subsingleton.ae_strongly_measurable' {m : MeasurableSpace α} [TopologicalSpace β]
[Subsingleton α] {μ : Measure α} (f : α → β) : AeStronglyMeasurable f μ :=
(Subsingleton.strongly_measurable' f).AeStronglyMeasurable
-#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aeStronglyMeasurable'
+#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.ae_strongly_measurable'
@[simp]
-theorem aeStronglyMeasurableZeroMeasure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
+theorem aeStronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
AeStronglyMeasurable f (0 : Measure α) :=
by
nontriviality α
inhabit α
exact ⟨fun x => f default, strongly_measurable_const, rfl⟩
-#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aeStronglyMeasurableZeroMeasure
+#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aeStronglyMeasurable_zero_measure
theorem SimpleFunc.aeStronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
(f : α →ₛ β) : AeStronglyMeasurable f μ :=
@@ -1257,11 +1258,11 @@ theorem ae_eq_mk (hf : AeStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AeStronglyMeasurable.ae_eq_mk
-protected theorem aeMeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
+protected theorem aEMeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AeStronglyMeasurable f μ) :
- AeMeasurable f μ :=
+ AEMeasurable f μ :=
⟨hf.mk f, hf.stronglyMeasurable_mk.Measurable, hf.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.ae_measurable MeasureTheory.AeStronglyMeasurable.aeMeasurable
+#align measure_theory.ae_strongly_measurable.ae_measurable MeasureTheory.AeStronglyMeasurable.aEMeasurable
end Mk
@@ -1274,24 +1275,24 @@ theorem aeStronglyMeasurable_congr (h : f =ᵐ[μ] g) :
⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩
#align ae_strongly_measurable_congr aeStronglyMeasurable_congr
-theorem monoMeasure {ν : Measure α} (hf : AeStronglyMeasurable f μ) (h : ν ≤ μ) :
+theorem mono_measure {ν : Measure α} (hf : AeStronglyMeasurable f μ) (h : ν ≤ μ) :
AeStronglyMeasurable f ν :=
⟨hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AeStronglyMeasurable.monoMeasure
+#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AeStronglyMeasurable.mono_measure
protected theorem mono' {ν : Measure α} (h : AeStronglyMeasurable f μ) (h' : ν ≪ μ) :
AeStronglyMeasurable f ν :=
⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AeStronglyMeasurable.mono'
-theorem monoSet {s t} (h : s ⊆ t) (ht : AeStronglyMeasurable f (μ.restrict t)) :
+theorem mono_set {s t} (h : s ⊆ t) (ht : AeStronglyMeasurable f (μ.restrict t)) :
AeStronglyMeasurable f (μ.restrict s) :=
- ht.monoMeasure (restrict_mono h le_rfl)
-#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AeStronglyMeasurable.monoSet
+ ht.mono_measure (restrict_mono h le_rfl)
+#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AeStronglyMeasurable.mono_set
protected theorem restrict (hfm : AeStronglyMeasurable f μ) {s} :
AeStronglyMeasurable f (μ.restrict s) :=
- hfm.monoMeasure Measure.restrict_le_self
+ hfm.mono_measure Measure.restrict_le_self
#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AeStronglyMeasurable.restrict
theorem ae_mem_imp_eq_mk {s} (h : AeStronglyMeasurable f (μ.restrict s)) :
@@ -1301,10 +1302,10 @@ theorem ae_mem_imp_eq_mk {s} (h : AeStronglyMeasurable f (μ.restrict s)) :
/-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable. -/
-theorem Continuous.compAeStronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
+theorem Continuous.comp_aeStronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
(hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => g (f x)) μ :=
⟨_, hg.comp_stronglyMeasurable hf.stronglyMeasurable_mk, EventuallyEq.fun_comp hf.ae_eq_mk g⟩
-#align continuous.comp_ae_strongly_measurable Continuous.compAeStronglyMeasurable
+#align continuous.comp_ae_strongly_measurable Continuous.comp_aeStronglyMeasurable
/-- A continuous function from `α` to `β` is ae strongly measurable when one of the two spaces is
second countable. -/
@@ -1314,11 +1315,11 @@ theorem Continuous.aeStronglyMeasurable [TopologicalSpace α] [OpensMeasurableSp
hf.StronglyMeasurable.AeStronglyMeasurable
#align continuous.ae_strongly_measurable Continuous.aeStronglyMeasurable
-protected theorem prodMk {f : α → β} {g : α → γ} (hf : AeStronglyMeasurable f μ)
+protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AeStronglyMeasurable f μ)
(hg : AeStronglyMeasurable g μ) : AeStronglyMeasurable (fun x => (f x, g x)) μ :=
⟨fun x => (hf.mk f x, hg.mk g x), hf.stronglyMeasurable_mk.prod_mk hg.stronglyMeasurable_mk,
hf.ae_eq_mk.prod_mk hg.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AeStronglyMeasurable.prodMk
+#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AeStronglyMeasurable.prod_mk
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
theorem Measurable.aeStronglyMeasurable {m : MeasurableSpace α} {μ : Measure α} [MeasurableSpace β]
@@ -1338,17 +1339,17 @@ protected theorem mul [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f
#align measure_theory.ae_strongly_measurable.add MeasureTheory.AeStronglyMeasurable.add
@[to_additive]
-protected theorem mulConst [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
+protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
AeStronglyMeasurable (fun x => f x * c) μ :=
- hf.mul aeStronglyMeasurableConst
-#align measure_theory.ae_strongly_measurable.mul_const MeasureTheory.AeStronglyMeasurable.mulConst
+ hf.mul aeStronglyMeasurable_const
+#align measure_theory.ae_strongly_measurable.mul_const MeasureTheory.AeStronglyMeasurable.mul_const
#align measure_theory.ae_strongly_measurable.add_const MeasureTheory.AeStronglyMeasurable.add_const
@[to_additive]
-protected theorem constMul [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
+protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AeStronglyMeasurable f μ) (c : β) :
AeStronglyMeasurable (fun x => c * f x) μ :=
- aeStronglyMeasurableConst.mul hf
-#align measure_theory.ae_strongly_measurable.const_mul MeasureTheory.AeStronglyMeasurable.constMul
+ aeStronglyMeasurable_const.mul hf
+#align measure_theory.ae_strongly_measurable.const_mul MeasureTheory.AeStronglyMeasurable.const_mul
#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AeStronglyMeasurable.const_add
@[to_additive]
@@ -1370,25 +1371,25 @@ protected theorem div [Group β] [TopologicalGroup β] (hf : AeStronglyMeasurabl
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
AeStronglyMeasurable (fun x => f x • g x) μ :=
- continuous_smul.compAeStronglyMeasurable (hf.prod_mk hg)
+ continuous_smul.comp_aeStronglyMeasurable (hf.prod_mk hg)
#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AeStronglyMeasurable.smul
#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AeStronglyMeasurable.vadd
-protected theorem constSmul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : AeStronglyMeasurable f μ)
- (c : 𝕜) : AeStronglyMeasurable (c • f) μ :=
+protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
+ (hf : AeStronglyMeasurable f μ) (c : 𝕜) : AeStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.stronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
-#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AeStronglyMeasurable.constSmul
+#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AeStronglyMeasurable.const_smul
-protected theorem constSmul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
+protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
(hf : AeStronglyMeasurable f μ) (c : 𝕜) : AeStronglyMeasurable (fun x => c • f x) μ :=
hf.const_smul c
-#align measure_theory.ae_strongly_measurable.const_smul' MeasureTheory.AeStronglyMeasurable.constSmul'
+#align measure_theory.ae_strongly_measurable.const_smul' MeasureTheory.AeStronglyMeasurable.const_smul'
@[to_additive]
-protected theorem smulConst {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
+protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
(hf : AeStronglyMeasurable f μ) (c : β) : AeStronglyMeasurable (fun x => f x • c) μ :=
- continuous_smul.compAeStronglyMeasurable (hf.prod_mk aeStronglyMeasurableConst)
-#align measure_theory.ae_strongly_measurable.smul_const MeasureTheory.AeStronglyMeasurable.smulConst
+ continuous_smul.comp_aeStronglyMeasurable (hf.prod_mk aeStronglyMeasurable_const)
+#align measure_theory.ae_strongly_measurable.smul_const MeasureTheory.AeStronglyMeasurable.smul_const
#align measure_theory.ae_strongly_measurable.vadd_const MeasureTheory.AeStronglyMeasurable.vadd_const
end Arithmetic
@@ -1419,22 +1420,22 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
@[to_additive]
-theorem List.aeStronglyMeasurableProd' (l : List (α → M)) (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) :
- AeStronglyMeasurable l.Prod μ :=
+theorem List.aeStronglyMeasurable_prod' (l : List (α → M))
+ (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) : AeStronglyMeasurable l.Prod μ :=
by
induction' l with f l ihl; · exact ae_strongly_measurable_one
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
-#align list.ae_strongly_measurable_prod' List.aeStronglyMeasurableProd'
-#align list.ae_strongly_measurable_sum' List.ae_strongly_measurable_sum'
+#align list.ae_strongly_measurable_prod' List.aeStronglyMeasurable_prod'
+#align list.ae_strongly_measurable_sum' List.aeStronglyMeasurable_sum'
@[to_additive]
-theorem List.aeStronglyMeasurableProd (l : List (α → M)) (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) :
+theorem List.aeStronglyMeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AeStronglyMeasurable f μ) :
AeStronglyMeasurable (fun x => (l.map fun f : α → M => f x).Prod) μ := by
simpa only [← Pi.list_prod_apply] using l.ae_strongly_measurable_prod' hl
-#align list.ae_strongly_measurable_prod List.aeStronglyMeasurableProd
-#align list.ae_strongly_measurable_sum List.ae_strongly_measurable_sum
+#align list.ae_strongly_measurable_prod List.aeStronglyMeasurable_prod
+#align list.ae_strongly_measurable_sum List.aeStronglyMeasurable_sum
end Monoid
@@ -1443,38 +1444,38 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
@[to_additive]
-theorem Multiset.aeStronglyMeasurableProd' (l : Multiset (α → M))
+theorem Multiset.aeStronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, AeStronglyMeasurable f μ) : AeStronglyMeasurable l.Prod μ :=
by
rcases l with ⟨l⟩
simpa using l.ae_strongly_measurable_prod' (by simpa using hl)
-#align multiset.ae_strongly_measurable_prod' Multiset.aeStronglyMeasurableProd'
-#align multiset.ae_strongly_measurable_sum' Multiset.ae_strongly_measurable_sum'
+#align multiset.ae_strongly_measurable_prod' Multiset.aeStronglyMeasurable_prod'
+#align multiset.ae_strongly_measurable_sum' Multiset.aeStronglyMeasurable_sum'
@[to_additive]
-theorem Multiset.aeStronglyMeasurableProd (s : Multiset (α → M))
+theorem Multiset.aeStronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, AeStronglyMeasurable f μ) :
AeStronglyMeasurable (fun x => (s.map fun f : α → M => f x).Prod) μ := by
simpa only [← Pi.multiset_prod_apply] using s.ae_strongly_measurable_prod' hs
-#align multiset.ae_strongly_measurable_prod Multiset.aeStronglyMeasurableProd
-#align multiset.ae_strongly_measurable_sum Multiset.ae_strongly_measurable_sum
+#align multiset.ae_strongly_measurable_prod Multiset.aeStronglyMeasurable_prod
+#align multiset.ae_strongly_measurable_sum Multiset.aeStronglyMeasurable_sum
@[to_additive]
-theorem Finset.aeStronglyMeasurableProd' {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem Finset.aeStronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AeStronglyMeasurable (f i) μ) : AeStronglyMeasurable (∏ i in s, f i) μ :=
- Multiset.aeStronglyMeasurableProd' _ fun g hg =>
+ Multiset.aeStronglyMeasurable_prod' _ fun g hg =>
let ⟨i, hi, hg⟩ := Multiset.mem_map.1 hg
hg ▸ hf _ hi
-#align finset.ae_strongly_measurable_prod' Finset.aeStronglyMeasurableProd'
-#align finset.ae_strongly_measurable_sum' Finset.ae_strongly_measurable_sum'
+#align finset.ae_strongly_measurable_prod' Finset.aeStronglyMeasurable_prod'
+#align finset.ae_strongly_measurable_sum' Finset.aeStronglyMeasurable_sum'
@[to_additive]
-theorem Finset.aeStronglyMeasurableProd {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem Finset.aeStronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AeStronglyMeasurable (f i) μ) :
AeStronglyMeasurable (fun a => ∏ i in s, f i a) μ := by
simpa only [← Finset.prod_apply] using s.ae_strongly_measurable_prod' hf
-#align finset.ae_strongly_measurable_prod Finset.aeStronglyMeasurableProd
-#align finset.ae_strongly_measurable_sum Finset.ae_strongly_measurable_sum
+#align finset.ae_strongly_measurable_prod Finset.aeStronglyMeasurable_prod
+#align finset.ae_strongly_measurable_sum Finset.aeStronglyMeasurable_sum
end CommMonoid
@@ -1483,56 +1484,56 @@ section SecondCountableAeStronglyMeasurable
variable [MeasurableSpace β]
/-- In a space with second countable topology, measurable implies strongly measurable. -/
-theorem AeMeasurable.aeStronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
- [SecondCountableTopology β] (hf : AeMeasurable f μ) : AeStronglyMeasurable f μ :=
+theorem AEMeasurable.aeStronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
+ [SecondCountableTopology β] (hf : AEMeasurable f μ) : AeStronglyMeasurable f μ :=
⟨hf.mk f, hf.measurable_mk.StronglyMeasurable, hf.ae_eq_mk⟩
-#align ae_measurable.ae_strongly_measurable AeMeasurable.aeStronglyMeasurable
+#align ae_measurable.ae_strongly_measurable AEMeasurable.aeStronglyMeasurable
-theorem aeStronglyMeasurableId {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
+theorem aeStronglyMeasurable_id {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
{m : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
AeStronglyMeasurable (id : α → α) μ :=
- aeMeasurableId.AeStronglyMeasurable
-#align ae_strongly_measurable_id aeStronglyMeasurableId
+ aemeasurable_id.AeStronglyMeasurable
+#align ae_strongly_measurable_id aeStronglyMeasurable_id
/-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
-theorem aeStronglyMeasurable_iff_aeMeasurable [PseudoMetrizableSpace β] [BorelSpace β]
- [SecondCountableTopology β] : AeStronglyMeasurable f μ ↔ AeMeasurable f μ :=
- ⟨fun h => h.AeMeasurable, fun h => h.AeStronglyMeasurable⟩
-#align ae_strongly_measurable_iff_ae_measurable aeStronglyMeasurable_iff_aeMeasurable
+theorem aeStronglyMeasurable_iff_aEMeasurable [PseudoMetrizableSpace β] [BorelSpace β]
+ [SecondCountableTopology β] : AeStronglyMeasurable f μ ↔ AEMeasurable f μ :=
+ ⟨fun h => h.AEMeasurable, fun h => h.AeStronglyMeasurable⟩
+#align ae_strongly_measurable_iff_ae_measurable aeStronglyMeasurable_iff_aEMeasurable
end SecondCountableAeStronglyMeasurable
protected theorem dist {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
AeStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
- continuous_dist.compAeStronglyMeasurable (hf.prod_mk hg)
+ continuous_dist.comp_aeStronglyMeasurable (hf.prod_mk hg)
#align measure_theory.ae_strongly_measurable.dist MeasureTheory.AeStronglyMeasurable.dist
protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => ‖f x‖) μ :=
- continuous_norm.compAeStronglyMeasurable hf
+ continuous_norm.comp_aeStronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AeStronglyMeasurable.norm
protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AeStronglyMeasurable f μ) : AeStronglyMeasurable (fun x => ‖f x‖₊) μ :=
- continuous_nnnorm.compAeStronglyMeasurable hf
+ continuous_nnnorm.comp_aeStronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AeStronglyMeasurable.nnnorm
protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
- (hf : AeStronglyMeasurable f μ) : AeMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
- (ENNReal.continuous_coe.compAeStronglyMeasurable hf.nnnorm).AeMeasurable
+ (hf : AeStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
+ (ENNReal.continuous_coe.comp_aeStronglyMeasurable hf.nnnorm).AEMeasurable
#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AeStronglyMeasurable.ennnorm
protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α → β}
(hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
- AeMeasurable (fun a => edist (f a) (g a)) μ :=
- (continuous_edist.compAeStronglyMeasurable (hf.prod_mk hg)).AeMeasurable
+ AEMeasurable (fun a => edist (f a) (g a)) μ :=
+ (continuous_edist.comp_aeStronglyMeasurable (hf.prod_mk hg)).AEMeasurable
#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AeStronglyMeasurable.edist
-protected theorem realToNnreal {f : α → ℝ} (hf : AeStronglyMeasurable f μ) :
+protected theorem real_toNNReal {f : α → ℝ} (hf : AeStronglyMeasurable f μ) :
AeStronglyMeasurable (fun x => (f x).toNNReal) μ :=
- continuous_real_toNNReal.compAeStronglyMeasurable hf
-#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AeStronglyMeasurable.realToNnreal
+ continuous_real_toNNReal.comp_aeStronglyMeasurable hf
+#align measure_theory.ae_strongly_measurable.real_to_nnreal MeasureTheory.AeStronglyMeasurable.real_toNNReal
theorem aeStronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
AeStronglyMeasurable (indicator s f) μ ↔ AeStronglyMeasurable f (μ.restrict s) :=
@@ -1554,7 +1555,7 @@ protected theorem indicator [Zero β] (hfm : AeStronglyMeasurable f μ) {s : Set
(aeStronglyMeasurable_indicator_iff hs).mpr hfm.restrict
#align measure_theory.ae_strongly_measurable.indicator MeasureTheory.AeStronglyMeasurable.indicator
-theorem nullMeasurableSetEqFun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
+theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
(hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
NullMeasurableSet { x | f x = g x } μ :=
by
@@ -1564,9 +1565,9 @@ theorem nullMeasurableSetEqFun {E} [TopologicalSpace E] [MetrizableSpace E] {f g
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x = hg.mk g x) = (f x = g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AeStronglyMeasurable.nullMeasurableSetEqFun
+#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_eq_fun
-theorem nullMeasurableSetLt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
+theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
NullMeasurableSet { a | f a < g a } μ :=
by
@@ -1575,9 +1576,9 @@ theorem nullMeasurableSetLt [LinearOrder β] [OrderClosedTopology β] [PseudoMet
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x < hg.mk g x) = (f x < g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AeStronglyMeasurable.nullMeasurableSetLt
+#align measure_theory.ae_strongly_measurable.null_measurable_set_lt MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_lt
-theorem nullMeasurableSetLe [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
+theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ :=
by
@@ -1586,32 +1587,32 @@ theorem nullMeasurableSetLe [Preorder β] [OrderClosedTopology β] [PseudoMetriz
filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk]with x hfx hgx
change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
simp only [hfx, hgx]
-#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AeStronglyMeasurable.nullMeasurableSetLe
+#align measure_theory.ae_strongly_measurable.null_measurable_set_le MeasureTheory.AeStronglyMeasurable.nullMeasurableSet_le
-theorem aeStronglyMeasurableOfAeStronglyMeasurableTrim {α} {m m0 : MeasurableSpace α}
+theorem aeStronglyMeasurable_of_aeStronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
{μ : Measure α} (hm : m ≤ m0) {f : α → β} (hf : AeStronglyMeasurable f (μ.trim hm)) :
AeStronglyMeasurable f μ :=
⟨hf.mk f, StronglyMeasurable.mono hf.stronglyMeasurable_mk hm, ae_eq_of_ae_eq_trim hf.ae_eq_mk⟩
-#align ae_strongly_measurable_of_ae_strongly_measurable_trim aeStronglyMeasurableOfAeStronglyMeasurableTrim
+#align ae_strongly_measurable_of_ae_strongly_measurable_trim aeStronglyMeasurable_of_aeStronglyMeasurable_trim
-theorem compAeMeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
- {μ : Measure γ} (hg : AeStronglyMeasurable g (Measure.map f μ)) (hf : AeMeasurable f μ) :
+theorem comp_aEMeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
+ {μ : Measure γ} (hg : AeStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
AeStronglyMeasurable (g ∘ f) μ :=
- ⟨hg.mk g ∘ hf.mk f, hg.stronglyMeasurable_mk.compMeasurable hf.measurable_mk,
+ ⟨hg.mk g ∘ hf.mk f, hg.stronglyMeasurable_mk.comp_measurable hf.measurable_mk,
(ae_eq_comp hf hg.ae_eq_mk).trans (hf.ae_eq_mk.fun_comp (hg.mk g))⟩
-#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AeStronglyMeasurable.compAeMeasurable
+#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AeStronglyMeasurable.comp_aEMeasurable
-theorem compMeasurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
+theorem comp_measurable {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AeStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
AeStronglyMeasurable (g ∘ f) μ :=
- hg.compAeMeasurable hf.AeMeasurable
-#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AeStronglyMeasurable.compMeasurable
+ hg.comp_aemeasurable hf.AEMeasurable
+#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AeStronglyMeasurable.comp_measurable
-theorem compQuasiMeasurePreserving {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α}
+theorem comp_quasiMeasurePreserving {γ : Type _} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α}
{f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AeStronglyMeasurable g ν)
(hf : QuasiMeasurePreserving f μ ν) : AeStronglyMeasurable (g ∘ f) μ :=
- (hg.mono' hf.AbsolutelyContinuous).compMeasurable hf.Measurable
-#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AeStronglyMeasurable.compQuasiMeasurePreserving
+ (hg.mono' hf.AbsolutelyContinuous).comp_measurable hf.Measurable
+#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AeStronglyMeasurable.comp_quasiMeasurePreserving
theorem isSeparable_ae_range (hf : AeStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
@@ -1623,11 +1624,11 @@ theorem isSeparable_ae_range (hf : AeStronglyMeasurable f μ) :
/-- A function is almost everywhere strongly measurable if and only if it is almost everywhere
measurable, and up to a zero measure set its range is contained in a separable set. -/
-theorem aeStronglyMeasurable_iff_aeMeasurable_separable [PseudoMetrizableSpace β]
+theorem aeStronglyMeasurable_iff_aEMeasurable_separable [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] :
- AeStronglyMeasurable f μ ↔ AeMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
+ AeStronglyMeasurable f μ ↔ AEMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
by
- refine' ⟨fun H => ⟨H.AeMeasurable, H.isSeparable_ae_range⟩, _⟩
+ refine' ⟨fun H => ⟨H.AEMeasurable, H.isSeparable_ae_range⟩, _⟩
rintro ⟨H, ⟨t, t_sep, ht⟩⟩
rcases eq_empty_or_nonempty t with (rfl | h₀)
· simp only [mem_empty_iff_false, eventually_false_iff_eq_bot, ae_eq_bot] at ht
@@ -1637,13 +1638,13 @@ theorem aeStronglyMeasurable_iff_aeMeasurable_separable [PseudoMetrizableSpace
H.exists_ae_eq_range_subset ht h₀
refine' ⟨g, _, fg⟩
exact stronglyMeasurable_iff_measurable_separable.2 ⟨g_meas, t_sep.mono GT.gt⟩
-#align ae_strongly_measurable_iff_ae_measurable_separable aeStronglyMeasurable_iff_aeMeasurable_separable
+#align ae_strongly_measurable_iff_ae_measurable_separable aeStronglyMeasurable_iff_aEMeasurable_separable
theorem MeasurableEmbedding.aeStronglyMeasurable_map_iff {γ : Type _} {mγ : MeasurableSpace γ}
{mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ} (hf : MeasurableEmbedding f) {g : α → β} :
AeStronglyMeasurable g (Measure.map f μ) ↔ AeStronglyMeasurable (g ∘ f) μ :=
by
- refine' ⟨fun H => H.compMeasurable hf.measurable, _⟩
+ refine' ⟨fun H => H.comp_measurable hf.measurable, _⟩
rintro ⟨g₁, hgm₁, heq⟩
rcases hf.exists_strongly_measurable_extend hgm₁ fun x => ⟨g x⟩ with ⟨g₂, hgm₂, rfl⟩
exact ⟨g₂, hgm₂, hf.ae_map_iff.2 HEq⟩
@@ -1656,7 +1657,7 @@ theorem Embedding.aeStronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
letI := pseudo_metrizable_space_pseudo_metric γ
borelize β γ
refine'
- ⟨fun H => aeStronglyMeasurable_iff_aeMeasurable_separable.2 ⟨_, _⟩, fun H =>
+ ⟨fun H => aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩, fun H =>
hg.continuous.comp_ae_strongly_measurable H⟩
· let G : β → range g := cod_restrict g (range g) mem_range_self
have hG : ClosedEmbedding G :=
@@ -1666,9 +1667,9 @@ theorem Embedding.aeStronglyMeasurable_comp_iff [PseudoMetrizableSpace β] [Pseu
apply eq_univ_of_forall
rintro ⟨-, ⟨x, rfl⟩⟩
exact mem_range_self x }
- have : AeMeasurable (G ∘ f) μ := AeMeasurable.subtypeMk H.ae_measurable
+ have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.ae_measurable
exact hG.measurable_embedding.ae_measurable_comp_iff.1 this
- · rcases(aeStronglyMeasurable_iff_aeMeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
+ · rcases(aeStronglyMeasurable_iff_aEMeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
exact ⟨g ⁻¹' t, hg.is_separable_preimage ht, h't⟩
#align embedding.ae_strongly_measurable_comp_iff Embedding.aeStronglyMeasurable_comp_iff
@@ -1681,16 +1682,16 @@ theorem MeasureTheory.MeasurePreserving.aeStronglyMeasurable_comp_iff {β : Type
/-- An almost everywhere sequential limit of almost everywhere strongly measurable functions is
almost everywhere strongly measurable. -/
-theorem aeStronglyMeasurableOfTendstoAe {ι : Type _} [PseudoMetrizableSpace β] (u : Filter ι)
+theorem aeStronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace β] (u : Filter ι)
[NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
(hf : ∀ i, AeStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
AeStronglyMeasurable g μ := by
borelize β
- refine' aeStronglyMeasurable_iff_aeMeasurable_separable.2 ⟨_, _⟩
- · exact aeMeasurableOfTendstoMetrizableAe _ (fun n => (hf n).AeMeasurable) limUnder
+ refine' aeStronglyMeasurable_iff_aEMeasurable_separable.2 ⟨_, _⟩
+ · exact aEMeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).AEMeasurable) limUnder
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, is_separable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
- (aeStronglyMeasurable_iff_aeMeasurable_separable.1 (hf (v n))).2
+ (aeStronglyMeasurable_iff_aEMeasurable_separable.1 (hf (v n))).2
choose t t_sep ht using this
refine' ⟨closure (⋃ i, t i), (is_separable_Union fun i => t_sep i).closure, _⟩
filter_upwards [ae_all_iff.2 ht, limUnder]with x hx h'x
@@ -1698,7 +1699,7 @@ theorem aeStronglyMeasurableOfTendstoAe {ι : Type _} [PseudoMetrizableSpace β]
apply eventually_of_forall fun n => _
apply mem_Union_of_mem n
exact hx n
-#align ae_strongly_measurable_of_tendsto_ae aeStronglyMeasurableOfTendstoAe
+#align ae_strongly_measurable_of_tendsto_ae aeStronglyMeasurable_of_tendsto_ae
/-- If a sequence of almost everywhere strongly measurable functions converges almost everywhere,
one can select a strongly measurable function as the almost everywhere limit. -/
@@ -1711,34 +1712,34 @@ theorem exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
borelize β
obtain ⟨g, g_meas, hg⟩ :
∃ (g : α → β)(g_meas : Measurable g), ∀ᵐ x ∂μ, tendsto (fun n => f n x) at_top (𝓝 (g x)) :=
- measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).AeMeasurable) h_ae_tendsto
- have Hg : ae_strongly_measurable g μ := aeStronglyMeasurableOfTendstoAe _ hf hg
+ measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).AEMeasurable) h_ae_tendsto
+ have Hg : ae_strongly_measurable g μ := aeStronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.strongly_measurable_mk, _⟩
filter_upwards [hg, Hg.ae_eq_mk]with x hx h'x
rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
-theorem sumMeasure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
+theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
(h : ∀ i, AeStronglyMeasurable f (μ i)) : AeStronglyMeasurable f (Measure.sum μ) :=
by
borelize β
refine'
- aeStronglyMeasurable_iff_aeMeasurable_separable.2
- ⟨AeMeasurable.sumMeasure fun i => (h i).AeMeasurable, _⟩
+ aeStronglyMeasurable_iff_aEMeasurable_separable.2
+ ⟨AEMeasurable.sum_measure fun i => (h i).AEMeasurable, _⟩
have A : ∀ i : ι, ∃ t : Set β, is_separable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
- (aeStronglyMeasurable_iff_aeMeasurable_separable.1 (h i)).2
+ (aeStronglyMeasurable_iff_aEMeasurable_separable.1 (h i)).2
choose t t_sep ht using A
refine' ⟨⋃ i, t i, is_separable_Union t_sep, _⟩
simp only [measure.ae_sum_eq, mem_Union, eventually_supr]
intro i
filter_upwards [ht i]with x hx
exact ⟨i, hx⟩
-#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AeStronglyMeasurable.sumMeasure
+#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AeStronglyMeasurable.sum_measure
@[simp]
theorem aeStronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β] {m : MeasurableSpace α}
{μ : ι → Measure α} : AeStronglyMeasurable f (Sum μ) ↔ ∀ i, AeStronglyMeasurable f (μ i) :=
- ⟨fun h i => h.monoMeasure (Measure.le_sum _ _), sumMeasure⟩
+ ⟨fun h i => h.mono_measure (Measure.le_sum _ _), sum_measure⟩
#align ae_strongly_measurable_sum_measure_iff aeStronglyMeasurable_sum_measure_iff
@[simp]
@@ -1749,23 +1750,24 @@ theorem aeStronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Me
rfl
#align ae_strongly_measurable_add_measure_iff aeStronglyMeasurable_add_measure_iff
-theorem addMeasure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
+theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
(hμ : AeStronglyMeasurable f μ) (hν : AeStronglyMeasurable f ν) :
AeStronglyMeasurable f (μ + ν) :=
aeStronglyMeasurable_add_measure_iff.2 ⟨hμ, hν⟩
-#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AeStronglyMeasurable.addMeasure
+#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AeStronglyMeasurable.add_measure
-protected theorem union [PseudoMetrizableSpace β] {s : ι → Set α}
+protected theorem unionᵢ [PseudoMetrizableSpace β] {s : ι → Set α}
(h : ∀ i, AeStronglyMeasurable f (μ.restrict (s i))) :
AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
- (sumMeasure h).monoMeasure <| restrict_unionᵢ_le
-#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AeStronglyMeasurable.union
+ (sum_measure h).mono_measure <| restrict_unionᵢ_le
+#align measure_theory.ae_strongly_measurable.Union MeasureTheory.AeStronglyMeasurable.unionᵢ
@[simp]
theorem aeStronglyMeasurable_unionᵢ_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
AeStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
∀ i, AeStronglyMeasurable f (μ.restrict (s i)) :=
- ⟨fun h i => h.monoMeasure <| restrict_mono (subset_unionᵢ _ _) le_rfl, AeStronglyMeasurable.union⟩
+ ⟨fun h i => h.mono_measure <| restrict_mono (subset_unionᵢ _ _) le_rfl,
+ AeStronglyMeasurable.unionᵢ⟩
#align ae_strongly_measurable_Union_iff aeStronglyMeasurable_unionᵢ_iff
@[simp]
@@ -1783,10 +1785,10 @@ theorem aeStronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
by rw [uIoc_eq_union, aeStronglyMeasurable_union_iff]
#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AeStronglyMeasurable.aeStronglyMeasurable_uIoc_iff
-theorem smulMeasure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
+theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(h : AeStronglyMeasurable f μ) (c : R) : AeStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
-#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AeStronglyMeasurable.smulMeasure
+#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AeStronglyMeasurable.smul_measure
section NormedSpace
@@ -1837,16 +1839,16 @@ theorem StronglyMeasurable.apply_continuousLinearMap {m : MeasurableSpace α} {
(ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_stronglyMeasurable hφ
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
-theorem applyContinuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AeStronglyMeasurable φ μ) (v : F) :
+theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AeStronglyMeasurable φ μ) (v : F) :
AeStronglyMeasurable (fun a => φ a v) μ :=
- (ContinuousLinearMap.apply 𝕜 E v).Continuous.compAeStronglyMeasurable hφ
-#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AeStronglyMeasurable.applyContinuousLinearMap
+ (ContinuousLinearMap.apply 𝕜 E v).Continuous.comp_aeStronglyMeasurable hφ
+#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AeStronglyMeasurable.apply_continuousLinearMap
-theorem ContinuousLinearMap.aeStronglyMeasurableComp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
+theorem ContinuousLinearMap.aeStronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AeStronglyMeasurable f μ) (hg : AeStronglyMeasurable g μ) :
AeStronglyMeasurable (fun x => L (f x) (g x)) μ :=
- L.continuous₂.compAeStronglyMeasurable <| hf.prod_mk hg
-#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aeStronglyMeasurableComp₂
+ L.continuous₂.comp_aeStronglyMeasurable <| hf.prod_mk hg
+#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aeStronglyMeasurable_comp₂
end ContinuousLinearMapNontriviallyNormedField
@@ -1896,20 +1898,20 @@ protected noncomputable def mk (f : α → β) (hf : AeFinStronglyMeasurable f
hf.some
#align measure_theory.ae_fin_strongly_measurable.mk MeasureTheory.AeFinStronglyMeasurable.mk
-theorem finStronglyMeasurableMk (hf : AeFinStronglyMeasurable f μ) :
+theorem finStronglyMeasurable_mk (hf : AeFinStronglyMeasurable f μ) :
FinStronglyMeasurable (hf.mk f) μ :=
hf.choose_spec.1
-#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AeFinStronglyMeasurable.finStronglyMeasurableMk
+#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AeFinStronglyMeasurable.finStronglyMeasurable_mk
theorem ae_eq_mk (hf : AeFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AeFinStronglyMeasurable.ae_eq_mk
-protected theorem aeMeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
+protected theorem aEMeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AeFinStronglyMeasurable f μ) :
- AeMeasurable f μ :=
- ⟨hf.mk f, hf.finStronglyMeasurableMk.Measurable, hf.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.ae_measurable MeasureTheory.AeFinStronglyMeasurable.aeMeasurable
+ AEMeasurable f μ :=
+ ⟨hf.mk f, hf.finStronglyMeasurable_mk.Measurable, hf.ae_eq_mk⟩
+#align measure_theory.ae_fin_strongly_measurable.ae_measurable MeasureTheory.AeFinStronglyMeasurable.aEMeasurable
end Mk
@@ -1917,32 +1919,32 @@ section Arithmetic
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f * g) μ :=
- ⟨hf.mk f * hg.mk g, hf.finStronglyMeasurableMk.mul hg.finStronglyMeasurableMk,
+ ⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AeFinStronglyMeasurable.mul
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f + g) μ :=
- ⟨hf.mk f + hg.mk g, hf.finStronglyMeasurableMk.add hg.finStronglyMeasurableMk,
+ ⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.add hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AeFinStronglyMeasurable.add
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AeFinStronglyMeasurable f μ) :
AeFinStronglyMeasurable (-f) μ :=
- ⟨-hf.mk f, hf.finStronglyMeasurableMk.neg, hf.ae_eq_mk.neg⟩
+ ⟨-hf.mk f, hf.finStronglyMeasurable_mk.neg, hf.ae_eq_mk.neg⟩
#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AeFinStronglyMeasurable.neg
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f - g) μ :=
- ⟨hf.mk f - hg.mk g, hf.finStronglyMeasurableMk.sub hg.finStronglyMeasurableMk,
+ ⟨hf.mk f - hg.mk g, hf.finStronglyMeasurable_mk.sub hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AeFinStronglyMeasurable.sub
-protected theorem constSmul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜] [DistribMulAction 𝕜 β]
- [ContinuousSMul 𝕜 β] (hf : AeFinStronglyMeasurable f μ) (c : 𝕜) :
+protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
+ [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AeFinStronglyMeasurable f μ) (c : 𝕜) :
AeFinStronglyMeasurable (c • f) μ :=
- ⟨c • hf.mk f, hf.finStronglyMeasurableMk.const_smul c, hf.ae_eq_mk.const_smul c⟩
-#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AeFinStronglyMeasurable.constSmul
+ ⟨c • hf.mk f, hf.finStronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
+#align measure_theory.ae_fin_strongly_measurable.const_smul MeasureTheory.AeFinStronglyMeasurable.const_smul
end Arithmetic
@@ -1952,13 +1954,13 @@ variable [Zero β]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f ⊔ g) μ :=
- ⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurableMk.sup hg.finStronglyMeasurableMk,
+ ⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AeFinStronglyMeasurable.sup
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f ⊓ g) μ :=
- ⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurableMk.inf hg.finStronglyMeasurableMk,
+ ⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.inf hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.inf MeasureTheory.AeFinStronglyMeasurable.inf
@@ -1992,10 +1994,10 @@ theorem ae_eq_zero_compl (hf : AeFinStronglyMeasurable f μ) :
hf.exists_set_sigmaFinite.choose_spec.2.1
#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AeFinStronglyMeasurable.ae_eq_zero_compl
-instance sigmaFiniteRestrict (hf : AeFinStronglyMeasurable f μ) :
+instance sigmaFinite_restrict (hf : AeFinStronglyMeasurable f μ) :
SigmaFinite (μ.restrict hf.sigmaFiniteSet) :=
hf.exists_set_sigmaFinite.choose_spec.2.2
-#align measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict MeasureTheory.AeFinStronglyMeasurable.sigmaFiniteRestrict
+#align measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict MeasureTheory.AeFinStronglyMeasurable.sigmaFinite_restrict
end AeFinStronglyMeasurable
@@ -2014,10 +2016,10 @@ theorem finStronglyMeasurable_iff_measurable {m0 : MeasurableSpace α} (μ : Mea
/-- In a space with second countable topology and a sigma-finite measure,
`ae_fin_strongly_measurable` and `ae_measurable` are equivalent. -/
-theorem aeFinStronglyMeasurable_iff_aeMeasurable {m0 : MeasurableSpace α} (μ : Measure α)
- [SigmaFinite μ] : AeFinStronglyMeasurable f μ ↔ AeMeasurable f μ := by
- simp_rw [ae_fin_strongly_measurable, AeMeasurable, fin_strongly_measurable_iff_measurable]
-#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aeFinStronglyMeasurable_iff_aeMeasurable
+theorem aeFinStronglyMeasurable_iff_aEMeasurable {m0 : MeasurableSpace α} (μ : Measure α)
+ [SigmaFinite μ] : AeFinStronglyMeasurable f μ ↔ AEMeasurable f μ := by
+ simp_rw [ae_fin_strongly_measurable, AEMeasurable, fin_strongly_measurable_iff_measurable]
+#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aeFinStronglyMeasurable_iff_aEMeasurable
end SecondCountableTopology
mathlib commit https://github.com/leanprover-community/mathlib/commit/39478763114722f0ec7613cb2f3f7701f9b86c8d
@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit d3af0609f6db8691dffdc3e1fb7feb7da72698f2
+! leanprover-community/mathlib commit 7317149f12f55affbc900fc873d0d422485122b9
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Analysis.NormedSpace.BoundedLinearMaps
import Mathbin.Topology.MetricSpace.Metrizable
+import Mathbin.MeasureTheory.Integral.Lebesgue
import Mathbin.MeasureTheory.Function.SimpleFuncDense
/-!
mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit a75898643b2d774cced9ae7c0b28c21663b99666
+! leanprover-community/mathlib commit d3af0609f6db8691dffdc3e1fb7feb7da72698f2
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -262,7 +262,7 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
‖hf.approxBounded c n x‖ ≤ c :=
by
simp only [strongly_measurable.approx_bounded, simple_func.coe_map, Function.comp_apply]
- refine' (norm_smul _ _).le.trans _
+ refine' (norm_smul_le _ _).trans _
by_cases h0 : ‖hf.approx n x‖ = 0
· simp only [h0, div_zero, min_eq_right, zero_le_one, norm_zero, MulZeroClass.mul_zero]
exact hc
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d
@@ -783,8 +783,7 @@ theorem stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpac
else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
measurableSet_fiber' := by
intro x
- convert
- (hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
+ convert(hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diffₓ hs)
ext1 y
simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212
@@ -264,7 +264,7 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
simp only [strongly_measurable.approx_bounded, simple_func.coe_map, Function.comp_apply]
refine' (norm_smul _ _).le.trans _
by_cases h0 : ‖hf.approx n x‖ = 0
- · simp only [h0, div_zero, min_eq_right, zero_le_one, norm_zero, mul_zero]
+ · simp only [h0, div_zero, min_eq_right, zero_le_one, norm_zero, MulZeroClass.mul_zero]
exact hc
cases le_total ‖hf.approx n x‖ c
· rw [min_eq_left _]
mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936
@@ -208,7 +208,7 @@ protected theorem tendsto_approx {m : MeasurableSpace α} (hf : StronglyMeasurab
/-- Similar to `strongly_measurable.approx`, but enforces that the norm of every function in the
sequence is less than `c` everywhere. If `‖f x‖ ≤ c` this sequence of simple functions verifies
`tendsto (λ n, hf.approx_bounded n x) at_top (𝓝 (f x))`. -/
-noncomputable def approxBounded {m : MeasurableSpace α} [HasNorm β] [SMul ℝ β]
+noncomputable def approxBounded {m : MeasurableSpace α} [Norm β] [SMul ℝ β]
(hf : StronglyMeasurable f) (c : ℝ) : ℕ → SimpleFunc α β := fun n =>
(hf.approx n).map fun x => min 1 (c / ‖x‖) • x
#align measure_theory.strongly_measurable.approx_bounded MeasureTheory.StronglyMeasurable.approxBounded
mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442
@@ -299,7 +299,7 @@ theorem stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
end BasicPropertiesInAnyTopologicalSpace
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » t) -/
theorem finStronglyMeasurableOfSetSigmaFinite [TopologicalSpace β] [Zero β] {m : MeasurableSpace α}
{μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α} (ht : MeasurableSet t)
(hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) : FinStronglyMeasurable f μ :=
@@ -907,9 +907,9 @@ theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorde
exact (hf.prod_mk hg).Measurable is_closed_le_prod.measurable_set
#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hf_zero : ∀ (x) (_ : x ∉ s), f x = 0) :
@@ -933,7 +933,7 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
exact tendsto_const_nhds
#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
/-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
on `s` is `m`-strongly-measurable, then `f` is also `m₂`-strongly-measurable. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211
@@ -522,13 +522,13 @@ open Filter
open Filter
-protected theorem sup [HasSup β] [ContinuousSup β] (hf : StronglyMeasurable f)
+protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
(hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
-protected theorem inf [HasInf β] [ContinuousInf β] (hf : StronglyMeasurable f)
+protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
(hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee
@@ -63,7 +63,7 @@ measurable functions, as a basis for the Bochner integral.
open MeasureTheory Filter TopologicalSpace Function Set MeasureTheory.Measure
-open Ennreal Topology MeasureTheory NNReal BigOperators
+open ENNReal Topology MeasureTheory NNReal BigOperators
/-- The typeclass `second_countable_topology_either α β` registers the fact that at least one of
the two spaces has second countable topology. This is the right assumption to ensure that continuous
@@ -851,7 +851,7 @@ protected theorem nnnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCo
protected theorem ennnorm {m : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
- (Ennreal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).Measurable
+ (ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).Measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
protected theorem real_toNNReal {m : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
@@ -1114,7 +1114,7 @@ protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasura
⟨fun n => hf.approx n + hg.approx n, fun n =>
(measure_mono (Function.support_add _ _)).trans_lt
((measure_union_le _ _).trans_lt
- (Ennreal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
+ (ENNReal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add
@@ -1132,7 +1132,7 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
⟨fun n => hf.approx n - hg.approx n, fun n =>
(measure_mono (Function.support_sub _ _)).trans_lt
((measure_union_le _ _).trans_lt
- (Ennreal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
+ (ENNReal.add_lt_top.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩)),
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
@@ -1520,7 +1520,7 @@ protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α →
protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AeStronglyMeasurable f μ) : AeMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
- (Ennreal.continuous_coe.compAeStronglyMeasurable hf.nnnorm).AeMeasurable
+ (ENNReal.continuous_coe.compAeStronglyMeasurable hf.nnnorm).AeMeasurable
#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AeStronglyMeasurable.ennnorm
protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α → β}
@@ -1863,7 +1863,7 @@ theorem aeStronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
· rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
filter_upwards [hg']with a ha h'a
- have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, Ennreal.coe_eq_zero] using h'a
+ have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, ENNReal.coe_eq_zero] using h'a
rw [ha this]
· filter_upwards [ae_restrict_mem A.compl]with x hx
simp only [Classical.not_not, mem_set_of_eq, mem_compl_iff] at hx
@@ -1873,7 +1873,7 @@ theorem aeStronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
rw [eventually_eq, ae_with_density_iff hf.coe_nnreal_ennreal]
filter_upwards [hg']with x hx h'x
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
- simp only [Ne.def, Ennreal.coe_eq_zero] at h'x
+ simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
#align ae_strongly_measurable_with_density_iff aeStronglyMeasurable_withDensity_iff
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -460,7 +460,7 @@ protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable
#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg
@[to_additive]
-protected theorem div [Div β] [HasContinuousDiv β] (hf : StronglyMeasurable f)
+protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f / g) :=
⟨fun n => hf.approx n / hg.approx n, fun x => (hf.tendsto_approx x).div' (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.div MeasureTheory.StronglyMeasurable.div
@@ -1127,7 +1127,7 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMe
exact hf.fin_support_approx n
#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg
-protected theorem sub [AddGroup β] [HasContinuousSub β] (hf : FinStronglyMeasurable f μ)
+protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ :=
⟨fun n => hf.approx n - hg.approx n, fun n =>
(measure_mono (Function.support_sub _ _)).trans_lt
@@ -1932,7 +1932,7 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AeFinStrongly
⟨-hf.mk f, hf.finStronglyMeasurableMk.neg, hf.ae_eq_mk.neg⟩
#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AeFinStronglyMeasurable.neg
-protected theorem sub [AddGroup β] [HasContinuousSub β] (hf : AeFinStronglyMeasurable f μ)
+protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AeFinStronglyMeasurable f μ)
(hg : AeFinStronglyMeasurable g μ) : AeFinStronglyMeasurable (f - g) μ :=
⟨hf.mk f - hg.mk g, hf.finStronglyMeasurableMk.sub hg.finStronglyMeasurableMk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
deprecated
attributeWhy these changes?
@@ -138,7 +138,7 @@ theorem StronglyMeasurable.of_finite [Finite α] {_ : MeasurableSpace α}
(f : α → β) : StronglyMeasurable f :=
⟨fun _ => SimpleFunc.ofFinite f, fun _ => tendsto_const_nhds⟩
-@[deprecated] -- Since 2024/02/05
+@[deprecated] -- Since 2024-02-05
alias stronglyMeasurable_of_fintype := StronglyMeasurable.of_finite
@[deprecated StronglyMeasurable.of_finite]
@@ -1899,7 +1899,7 @@ theorem _root_.aestronglyMeasurable_withDensity_iff {E : Type*} [NormedAddCommGr
· rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
filter_upwards [hg'] with a ha h'a
- have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, ENNReal.coe_eq_zero] using h'a
+ have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h'a
rw [ha this]
· filter_upwards [ae_restrict_mem A.compl] with x hx
simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
@@ -1909,8 +1909,8 @@ theorem _root_.aestronglyMeasurable_withDensity_iff {E : Type*} [NormedAddCommGr
rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal]
filter_upwards [hg'] with x hx h'x
rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul]
- simp only [Ne.def, ENNReal.coe_eq_zero] at h'x
- simpa only [NNReal.coe_eq_zero, Ne.def] using h'x
+ simp only [Ne, ENNReal.coe_eq_zero] at h'x
+ simpa only [NNReal.coe_eq_zero, Ne] using h'x
#align ae_strongly_measurable_with_density_iff aestronglyMeasurable_withDensity_iff
end AEStronglyMeasurable
by volume_tac
to get a default measure in AEStronglyMeasurable
(#11771)
This is already the case for Integrable
and AEMeasurable
.
@@ -83,19 +83,22 @@ scoped notation "StronglyMeasurable[" m "]" => @MeasureTheory.StronglyMeasurable
/-- A function is `FinStronglyMeasurable` with respect to a measure if it is the limit of simple
functions with support with finite measure. -/
-def FinStronglyMeasurable [Zero β] {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
+def FinStronglyMeasurable [Zero β]
+ {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop :=
∃ fs : ℕ → α →ₛ β, (∀ n, μ (support (fs n)) < ∞) ∧ ∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
#align measure_theory.fin_strongly_measurable MeasureTheory.FinStronglyMeasurable
/-- A function is `AEStronglyMeasurable` with respect to a measure `μ` if it is almost everywhere
equal to the limit of a sequence of simple functions. -/
-def AEStronglyMeasurable {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
+def AEStronglyMeasurable
+ {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop :=
∃ g, StronglyMeasurable g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable MeasureTheory.AEStronglyMeasurable
/-- A function is `AEFinStronglyMeasurable` with respect to a measure if it is almost everywhere
equal to the limit of a sequence of simple functions with support with finite measure. -/
-def AEFinStronglyMeasurable [Zero β] {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
+def AEFinStronglyMeasurable
+ [Zero β] {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop :=
∃ g, FinStronglyMeasurable g μ ∧ f =ᵐ[μ] g
#align measure_theory.ae_fin_strongly_measurable MeasureTheory.AEFinStronglyMeasurable
@@ -1296,6 +1299,15 @@ protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasur
hf.ae_eq_mk.prod_mk hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk
+/-- The composition of a continuous function of two variables and two ae strongly measurable
+functions is ae strongly measurable. -/
+theorem _root_.Continuous.comp_aestronglyMeasurable₂
+ {β' : Type*} [TopologicalSpace β']
+ {g : β → β' → γ} {f : α → β} {f' : α → β'} (hg : Continuous g.uncurry)
+ (hf : AEStronglyMeasurable f μ) (h'f : AEStronglyMeasurable f' μ) :
+ AEStronglyMeasurable (fun x => g (f x) (f' x)) μ :=
+ hg.comp_aestronglyMeasurable (hf.prod_mk h'f)
+
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
@[aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α}
@@ -1870,7 +1882,7 @@ theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStrongly
theorem _root_.ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => L (f x) (g x)) μ :=
- L.continuous₂.comp_aestronglyMeasurable <| hf.prod_mk hg
+ L.continuous₂.comp_aestronglyMeasurable₂ hf hg
#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂
end ContinuousLinearMapNontriviallyNormedField
filter_upwards
(#11208)
This is presumably not exhaustive, but covers about a hundred instances.
Style opinions (e.g., why a particular change is great/not a good idea) are very welcome; I'm still forming my own.
@@ -641,7 +641,7 @@ protected theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace
apply this.mono
rintro _ ⟨x, rfl⟩
apply mem_closure_of_tendsto (hf.tendsto_approx x)
- refine eventually_of_forall fun n => ?_
+ filter_upwards with n
apply mem_iUnion_of_mem n
exact mem_range_self _
#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_range
@@ -776,7 +776,7 @@ theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace
rintro _ ⟨x, rfl⟩
rw [tendsto_pi_nhds] at lim
apply mem_closure_of_tendsto ((lim x).comp hv)
- refine eventually_of_forall fun n => ?_
+ filter_upwards with n
apply mem_iUnion_of_mem n
exact mem_range_self _
#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendsto
@@ -1697,9 +1697,7 @@ theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizable
refine ⟨closure (⋃ i, t i), .closure <| .iUnion t_sep, ?_⟩
filter_upwards [ae_all_iff.2 ht, lim] with x hx h'x
apply mem_closure_of_tendsto (h'x.comp hv)
- refine eventually_of_forall fun n => ?_
- apply mem_iUnion_of_mem n
- exact hx n
+ filter_upwards with n using mem_iUnion_of_mem n (hx n)
#align ae_strongly_measurable_of_tendsto_ae aestronglyMeasurable_of_tendsto_ae
/-- If a sequence of almost everywhere strongly measurable functions converges almost everywhere,
All these lemmas refer to the range of some function being open/range (i.e. isOpen
or isClosed
).
@@ -752,7 +752,7 @@ theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [T
· let G : β → range g := rangeFactorization g
have hG : ClosedEmbedding G :=
{ hg.codRestrict _ _ with
- closed_range := by
+ isClosed_range := by
rw [surjective_onto_range.range_eq]
exact isClosed_univ }
have : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
@@ -1667,7 +1667,7 @@ theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β
· let G : β → range g := rangeFactorization g
have hG : ClosedEmbedding G :=
{ hg.codRestrict _ _ with
- closed_range := by rw [surjective_onto_range.range_eq]; exact isClosed_univ }
+ isClosed_range := by rw [surjective_onto_range.range_eq]; exact isClosed_univ }
have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.aemeasurable
exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this
· rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -1816,7 +1816,6 @@ theorem smul_measure {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsS
section NormedSpace
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
-
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
theorem _root_.aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
@@ -1854,11 +1853,8 @@ end MulAction
section ContinuousLinearMapNontriviallyNormedField
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
theorem _root_.StronglyMeasurable.apply_continuousLinearMap {_m : MeasurableSpace α}
This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0
branch as we update to intermediate nightlies.
Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>
@@ -321,7 +321,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
rw [← Set.mem_iUnion, iUnion_spanningSets (μ.restrict t)]
trivial
refine' ⟨n, fun m hnm => _⟩
- simp_rw [SimpleFunc.restrict_apply _ ((hS_meas m).inter ht),
+ simp_rw [fs, SimpleFunc.restrict_apply _ ((hS_meas m).inter ht),
Set.indicator_of_mem (hn m hnm)]
rw [tendsto_atTop'] at h ⊢
intro s hs
@@ -1048,7 +1048,7 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
· have h_fs_zero : ∀ n, ∀ x ∈ tᶜ, fs n x = 0 := by
intro n x hxt
rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt
- simpa using hxt n
+ simpa [T] using hxt n
refine' fun x hxt => tendsto_nhds_unique (h_approx x) _
rw [funext fun n => h_fs_zero n x hxt]
exact tendsto_const_nhds
@@ -2099,7 +2099,7 @@ theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type*} [Topol
(fun p : ↥(t_sf n).range × α => u p.fst p.snd) ∘ fun p : ι × α =>
(⟨t_sf n p.fst, SimpleFunc.mem_range_self _ _⟩, p.snd) :=
rfl
- simp_rw [this]
+ simp_rw [U, this]
refine' h_meas.comp (Measurable.prod_mk _ measurable_snd)
exact ((t_sf n).measurable.comp measurable_fst).subtype_mk
#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurable
@@ -2139,7 +2139,7 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
(fun p : ↥(t_sf n).range × α => u p.fst p.snd) ∘ fun p : ι × α =>
(⟨t_sf n p.fst, SimpleFunc.mem_range_self _ _⟩, p.snd) :=
rfl
- simp_rw [this]
+ simp_rw [U, this]
refine' h_str_meas.comp_measurable (Measurable.prod_mk _ measurable_snd)
exact ((t_sf n).measurable.comp measurable_fst).subtype_mk
#align measure_theory.strongly_measurable_uncurry_of_continuous_of_strongly_measurable MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable
Homogenises porting notes via capitalisation and addition of whitespace.
It makes the following changes:
@@ -166,7 +166,7 @@ theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalS
exact funext fun x => hf x default
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
--- porting note: changed binding type of `MeasurableSpace α`.
+-- Porting note: changed binding type of `MeasurableSpace α`.
@[simp]
theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
[Subsingleton α] (f : α → β) : StronglyMeasurable f :=
@@ -105,7 +105,7 @@ open MeasureTheory
/-! ## Strongly measurable functions -/
-@[aesop 30% apply (rule_sets [Measurable])]
+@[aesop 30% apply (rule_sets := [Measurable])]
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {_ : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
AEStronglyMeasurable f μ :=
@@ -333,7 +333,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
/-- If the measure is sigma-finite, all strongly measurable functions are
`FinStronglyMeasurable`. -/
-@[aesop 5% apply (rule_sets [Measurable])]
+@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
(hf : StronglyMeasurable f) (μ : Measure α) [SigmaFinite μ] : FinStronglyMeasurable f μ :=
hf.finStronglyMeasurable_of_set_sigmaFinite MeasurableSet.univ (by simp)
@@ -341,7 +341,7 @@ protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : Me
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
/-- A strongly measurable function is measurable. -/
-@[aesop 5% apply (rule_sets [Measurable])]
+@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f :=
measurable_of_tendsto_metrizable (fun n => (hf.approx n).measurable)
@@ -349,7 +349,7 @@ protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [Pse
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
/-- A strongly measurable function is almost everywhere measurable. -/
-@[aesop 5% apply (rule_sets [Measurable])]
+@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
(hf : StronglyMeasurable f) : AEMeasurable f μ :=
@@ -409,7 +409,7 @@ section Arithmetic
variable {mα : MeasurableSpace α} [TopologicalSpace β]
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f * g) :=
⟨fun n => hf.approx n * hg.approx n, fun x => (hf.tendsto_approx x).mul (hg.tendsto_approx x)⟩
@@ -430,7 +430,7 @@ theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c :
#align measure_theory.strongly_measurable.const_mul MeasureTheory.StronglyMeasurable.const_mul
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable])) const_nsmul]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
StronglyMeasurable (f ^ n) :=
⟨fun k => hf.approx k ^ n, fun x => (hf.tendsto_approx x).pow n⟩
@@ -442,14 +442,14 @@ protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv
#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f / g) :=
⟨fun n => hf.approx n / hg.approx n, fun x => (hf.tendsto_approx x).div' (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.div MeasureTheory.StronglyMeasurable.div
#align measure_theory.strongly_measurable.sub MeasureTheory.StronglyMeasurable.sub
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => f x • g x :=
@@ -552,14 +552,14 @@ open Filter
open Filter
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
(hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
@@ -658,7 +658,7 @@ section SecondCountableStronglyMeasurable
variable {mα : MeasurableSpace α} [MeasurableSpace β]
/-- In a space with second countable topology, measurable implies strongly measurable. -/
-@[aesop 90% apply (rule_sets [Measurable])]
+@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
[SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
StronglyMeasurable f := by
@@ -860,7 +860,7 @@ protected theorem indicator {_ : MeasurableSpace α} [TopologicalSpace β] [Zero
hf.piecewise hs stronglyMeasurable_const
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
-@[aesop safe 20 apply (rule_sets [Measurable])]
+@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {_ : MeasurableSpace α} {β : Type*} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
@@ -1031,7 +1031,7 @@ protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop
end sequence
/-- A finitely strongly measurable function is strongly measurable. -/
-@[aesop 5% apply (rule_sets [Measurable])]
+@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
(hf : FinStronglyMeasurable f μ) : StronglyMeasurable f :=
⟨hf.approx, hf.tendsto_approx⟩
@@ -1070,7 +1070,7 @@ section Arithmetic
variable [TopologicalSpace β]
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ := by
refine'
@@ -1080,7 +1080,7 @@ protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMe
exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ :=
⟨fun n => hf.approx n + hg.approx n, fun n =>
@@ -1124,7 +1124,7 @@ section Order
variable [TopologicalSpace β] [Zero β]
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ := by
refine'
@@ -1134,7 +1134,7 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ := by
refine'
@@ -1230,7 +1230,7 @@ theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk
-@[aesop 5% apply (rule_sets [Measurable])]
+@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEStronglyMeasurable f μ) :
AEMeasurable f μ :=
@@ -1297,7 +1297,7 @@ protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasur
#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk
/-- In a space with second countable topology, measurable implies ae strongly measurable. -/
-@[aesop unsafe 30% apply (rule_sets [Measurable])]
+@[aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α}
[MeasurableSpace β] [PseudoMetrizableSpace β] [SecondCountableTopology β]
[OpensMeasurableSpace β] (hf : Measurable f) : AEStronglyMeasurable f μ :=
@@ -1306,7 +1306,7 @@ theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Me
section Arithmetic
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.stronglyMeasurable_mk.mul hg.stronglyMeasurable_mk,
@@ -1335,7 +1335,7 @@ protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f
#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AEStronglyMeasurable.neg
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ :=
⟨hf.mk f / hg.mk g, hf.stronglyMeasurable_mk.div hg.stronglyMeasurable_mk,
@@ -1343,7 +1343,7 @@ protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurabl
#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.div
#align measure_theory.ae_strongly_measurable.sub MeasureTheory.AEStronglyMeasurable.sub
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => f x • g x) μ :=
@@ -1351,7 +1351,7 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AEStronglyMeasurable.smul
#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AEStronglyMeasurable.vadd
-@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable])) const_nsmul]
+@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
AEStronglyMeasurable (f ^ n) μ :=
⟨hf.mk f ^ n, hf.stronglyMeasurable_mk.pow _, hf.ae_eq_mk.pow_const _⟩
@@ -1379,14 +1379,14 @@ end Arithmetic
section Order
-@[aesop safe 20 apply (rule_sets [Measurable])]
+@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.stronglyMeasurable_mk.sup hg.stronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.sup
-@[aesop safe 20 apply (rule_sets [Measurable])]
+@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.stronglyMeasurable_mk.inf hg.stronglyMeasurable_mk,
@@ -1468,7 +1468,7 @@ section SecondCountableAEStronglyMeasurable
variable [MeasurableSpace β]
/-- In a space with second countable topology, measurable implies strongly measurable. -/
-@[aesop 90% apply (rule_sets [Measurable])]
+@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
[SecondCountableTopology β] (hf : AEMeasurable f μ) : AEStronglyMeasurable f μ :=
⟨hf.mk f, hf.measurable_mk.stronglyMeasurable, hf.ae_eq_mk⟩
@@ -1489,7 +1489,7 @@ theorem _root_.aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β]
end SecondCountableAEStronglyMeasurable
-@[aesop safe 20 apply (rule_sets [Measurable])]
+@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {β : Type*} [PseudoMetricSpace β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
@@ -1514,7 +1514,7 @@ protected theorem ennnorm {β : Type*} [SeminormedAddCommGroup β] {f : α →
(ENNReal.continuous_coe.comp_aestronglyMeasurable hf.nnnorm).aemeasurable
#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AEStronglyMeasurable.ennnorm
-@[aesop safe 20 apply (rule_sets [Measurable])]
+@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem edist {β : Type*} [SeminormedAddCommGroup β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEMeasurable (fun a => edist (f a) (g a)) μ :=
@@ -1935,7 +1935,7 @@ theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
-@[aesop 10% apply (rule_sets [Measurable])]
+@[aesop 10% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
AEMeasurable f μ :=
@@ -1946,14 +1946,14 @@ end Mk
section Arithmetic
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mul
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ :=
⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
@@ -1986,14 +1986,14 @@ section Order
variable [Zero β]
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.sup
-@[aesop safe 20 (rule_sets [Measurable])]
+@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
@@ -2051,7 +2051,7 @@ theorem finStronglyMeasurable_iff_measurable {_m0 : MeasurableSpace α} (μ : Me
/-- In a space with second countable topology and a sigma-finite measure, a measurable function
is `FinStronglyMeasurable`. -/
-@[aesop 90% apply (rule_sets [Measurable])]
+@[aesop 90% apply (rule_sets := [Measurable])]
theorem finStronglyMeasurable_of_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
[SigmaFinite μ] (hf : Measurable f) : FinStronglyMeasurable f μ :=
(finStronglyMeasurable_iff_measurable μ).mpr hf
@@ -2065,7 +2065,7 @@ theorem aefinStronglyMeasurable_iff_aemeasurable {_m0 : MeasurableSpace α} (μ
/-- In a space with second countable topology and a sigma-finite measure,
an `AEMeasurable` function is `AEFinStronglyMeasurable`. -/
-@[aesop 90% apply (rule_sets [Measurable])]
+@[aesop 90% apply (rule_sets := [Measurable])]
theorem aefinStronglyMeasurable_of_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
[SigmaFinite μ] (hf : AEMeasurable f μ) : AEFinStronglyMeasurable f μ :=
(aefinStronglyMeasurable_iff_aemeasurable μ).mpr hf
refine
s (#10762)
I replaced a few "terminal" refine/refine'
s with exact
.
The strategy was very simple-minded: essentially any refine
whose following line had smaller indentation got replaced by exact
and then I cleaned up the mess.
This PR certainly leaves some further terminal refine
s, but maybe the current change is beneficial.
@@ -231,7 +231,7 @@ theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGrou
exact hfx
nth_rw 2 [this.symm]
refine' Tendsto.min tendsto_const_nhds _
- refine' Tendsto.div tendsto_const_nhds h_tendsto.norm hfx0
+ exact Tendsto.div tendsto_const_nhds h_tendsto.norm hfx0
#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
Rename AEStronglyMeasurable.mono'
to AEStronglyMeasurable.mono_ac
.
Partly forward-port https://github.com/leanprover-community/mathlib/pull/18863
Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
@@ -8,7 +8,7 @@ import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFuncDense
import Mathlib.Topology.Algebra.Module.FiniteDimension
-#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
+#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
/-!
# Strongly measurable and finitely strongly measurable functions
@@ -1206,7 +1206,7 @@ theorem SimpleFunc.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure
namespace AEStronglyMeasurable
-variable {m : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] [TopologicalSpace γ]
+variable {m : MeasurableSpace α} {μ ν : Measure α} [TopologicalSpace β] [TopologicalSpace γ]
{f g : α → β}
section Mk
@@ -1253,10 +1253,12 @@ theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν
⟨hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure
-protected theorem mono' {ν : Measure α} (h : AEStronglyMeasurable f μ) (h' : ν ≪ μ) :
- AEStronglyMeasurable f ν :=
- ⟨h.mk f, h.stronglyMeasurable_mk, h' h.ae_eq_mk⟩
-#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono'
+protected lemma mono_ac (h : ν ≪ μ) (hμ : AEStronglyMeasurable f μ) : AEStronglyMeasurable f ν :=
+ let ⟨g, hg, hg'⟩ := hμ; ⟨g, hg, h.ae_eq hg'⟩
+#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono_ac
+#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.AEStronglyMeasurable.mono_ac
+
+@[deprecated] protected alias mono' := AEStronglyMeasurable.mono_ac
theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
AEStronglyMeasurable f (μ.restrict s) :=
Fintype
-> Finite
(#10289)
hasFiniteIntegral_of_fintype
to HasFiniteIntegral.of_finite
, generalize to [Finite α]
, golf.integrable_of_fintype
to Integrable.of_finite
, generalize to [Finite α]
.SimpleFunc.ofFintype
to SimpleFunc.ofFinite
, generalize to [Finite α]
, use it to golf SimpleFunc.ofIsEmpty
.stronglyMeasurable_of_fintype
to StronglyMeasurable.of_finite
, deprecate stronglyMeasurable_of_isEmpty
.@@ -129,15 +129,20 @@ theorem SimpleFunc.stronglyMeasurable {α β} {_ : MeasurableSpace α} [Topologi
⟨fun _ => f, fun _ => tendsto_const_nhds⟩
#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable
-theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [TopologicalSpace β]
+@[nontriviality]
+theorem StronglyMeasurable.of_finite [Finite α] {_ : MeasurableSpace α}
+ [MeasurableSingletonClass α] [TopologicalSpace β]
(f : α → β) : StronglyMeasurable f :=
- ⟨fun _ => SimpleFunc.ofIsEmpty, isEmptyElim⟩
-#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
+ ⟨fun _ => SimpleFunc.ofFinite f, fun _ => tendsto_const_nhds⟩
-theorem stronglyMeasurable_of_fintype [Fintype α] {_ : MeasurableSpace α}
- [MeasurableSingletonClass α] [TopologicalSpace β]
+@[deprecated] -- Since 2024/02/05
+alias stronglyMeasurable_of_fintype := StronglyMeasurable.of_finite
+
+@[deprecated StronglyMeasurable.of_finite]
+theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [TopologicalSpace β]
(f : α → β) : StronglyMeasurable f :=
- ⟨fun _ => SimpleFunc.ofFintype f, fun _ => tendsto_const_nhds⟩
+ .of_finite f
+#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.StronglyMeasurable.of_finite
theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSpace β] {b : β} :
StronglyMeasurable fun _ : α => b :=
@@ -147,7 +152,7 @@ theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSp
@[to_additive]
theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpace β] [One β] :
StronglyMeasurable (1 : α → β) :=
- @stronglyMeasurable_const _ _ _ _ 1
+ stronglyMeasurable_const
#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_one
#align measure_theory.strongly_measurable_zero MeasureTheory.stronglyMeasurable_zero
@@ -155,10 +160,10 @@ theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpac
This version works for functions between empty types. -/
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
(hf : ∀ x y, f x = f y) : StronglyMeasurable f := by
- cases' isEmpty_or_nonempty α with _ h
- · exact stronglyMeasurable_of_isEmpty f
- · convert stronglyMeasurable_const (β := β) using 1
- exact funext fun x => hf x h.some
+ nontriviality α
+ inhabit α
+ convert stronglyMeasurable_const (β := β) using 1
+ exact funext fun x => hf x default
#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'
-- porting note: changed binding type of `MeasurableSpace α`.
IsSeparable
API (#10286)
isSeparable_iUnion
to an Iff
lemma, restore the original version as IsSeparable.iUnion
;isSeparable_union
and isSeparable_closure
;isSeparable_pi
from [Finite ι]
to [Countable ι]
, add IsSeparable.univ_pi
version;Dense.isSeparable_iff
and isSeparable_range
;isSeparable_of_separableSpace_subtype
to IsSeparable.of_subtype
;isSeparable_of_separableSpace
to IsSeparable.of_separableSpace
.@@ -629,10 +629,10 @@ theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M}
end CommMonoid
/-- The range of a strongly measurable function is separable. -/
-theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β] (hf : StronglyMeasurable f) :
- TopologicalSpace.IsSeparable (range f) := by
+protected theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β]
+ (hf : StronglyMeasurable f) : TopologicalSpace.IsSeparable (range f) := by
have : IsSeparable (closure (⋃ n, range (hf.approx n))) :=
- (isSeparable_iUnion fun n => (SimpleFunc.finite_range (hf.approx n)).isSeparable).closure
+ .closure <| .iUnion fun n => (hf.approx n).finite_range.isSeparable
apply this.mono
rintro _ ⟨x, rfl⟩
apply mem_closure_of_tendsto (hf.tendsto_approx x)
@@ -698,8 +698,7 @@ theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSp
cases h.out
· rw [stronglyMeasurable_iff_measurable_separable]
refine' ⟨hf.measurable, _⟩
- rw [← image_univ]
- exact (isSeparable_of_separableSpace univ).image hf
+ exact isSeparable_range hf
· exact hf.measurable.stronglyMeasurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
@@ -767,7 +766,7 @@ theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace
· exact measurable_of_tendsto_metrizable' u (fun i => (hf i).measurable) lim
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : IsSeparable (closure (⋃ i, range (f (v i)))) :=
- (isSeparable_iUnion fun i => (hf (v i)).isSeparable_range).closure
+ .closure <| .iUnion fun i => (hf (v i)).isSeparable_range
apply this.mono
rintro _ ⟨x, rfl⟩
rw [tendsto_pi_nhds] at lim
@@ -1688,7 +1687,7 @@ theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizable
have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
(aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
choose t t_sep ht using this
- refine' ⟨closure (⋃ i, t i), (isSeparable_iUnion fun i => t_sep i).closure, _⟩
+ refine ⟨closure (⋃ i, t i), .closure <| .iUnion t_sep, ?_⟩
filter_upwards [ae_all_iff.2 ht, lim] with x hx h'x
apply mem_closure_of_tendsto (h'x.comp hv)
refine eventually_of_forall fun n => ?_
@@ -1743,7 +1742,7 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
have A : ∀ i : ι, ∃ t : Set β, IsSeparable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
(aestronglyMeasurable_iff_aemeasurable_separable.1 (h i)).2
choose t t_sep ht using A
- refine' ⟨⋃ i, t i, isSeparable_iUnion t_sep, _⟩
+ refine ⟨⋃ i, t i, .iUnion t_sep, ?_⟩
simp only [Measure.ae_sum_eq, mem_iUnion, eventually_iSup]
intro i
filter_upwards [ht i] with x hx
@@ -2123,7 +2122,7 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
rw [this, measurable_swap_iff]
exact measurable_from_prod_countable fun j => (h j).measurable
· have : IsSeparable (⋃ i : (t_sf n).range, range (u i)) :=
- isSeparable_iUnion fun i => (h i).isSeparable_range
+ .iUnion fun i => (h i).isSeparable_range
apply this.mono
rintro _ ⟨⟨i, x⟩, rfl⟩
simp only [mem_iUnion, mem_range]
@@ -1111,7 +1111,7 @@ protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Mono
FinStronglyMeasurable (c • f) μ := by
refine' ⟨fun n => c • hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c⟩
rw [SimpleFunc.coe_smul]
- refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n)
+ exact (measure_mono (support_const_smul_subset c _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smul
end Arithmetic
@@ -1658,14 +1658,10 @@ theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β
refine'
⟨fun H => aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨_, _⟩, fun H =>
hg.continuous.comp_aestronglyMeasurable H⟩
- · let G : β → range g := codRestrict g (range g) mem_range_self
+ · let G : β → range g := rangeFactorization g
have hG : ClosedEmbedding G :=
{ hg.codRestrict _ _ with
- closed_range := by
- convert isClosed_univ (α := ↥(range g))
- apply eq_univ_of_forall
- rintro ⟨-, ⟨x, rfl⟩⟩
- exact mem_range_self x }
+ closed_range := by rw [surjective_onto_range.range_eq]; exact isClosed_univ }
have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.aemeasurable
exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this
· rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
@@ -3,10 +3,10 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
-import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFuncDense
+import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
This also adds some missing to_additive
s for vadd
.
@@ -425,6 +425,11 @@ theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c :
#align measure_theory.strongly_measurable.const_mul MeasureTheory.StronglyMeasurable.const_mul
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable])) const_nsmul]
+protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
+ StronglyMeasurable (f ^ n) :=
+ ⟨fun k => hf.approx k ^ n, fun x => (hf.tendsto_approx x).pow n⟩
+
@[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
@@ -447,13 +452,13 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
#align measure_theory.strongly_measurable.smul MeasureTheory.StronglyMeasurable.smul
#align measure_theory.strongly_measurable.vadd MeasureTheory.StronglyMeasurable.vadd
-@[measurability]
+@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable (c • f) :=
⟨fun n => c • hf.approx n, fun x => (hf.tendsto_approx x).const_smul c⟩
#align measure_theory.strongly_measurable.const_smul MeasureTheory.StronglyMeasurable.const_smul
-@[measurability]
+@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable fun x => c • f x :=
hf.const_smul c
@@ -1340,13 +1345,18 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AEStronglyMeasurable.smul
#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AEStronglyMeasurable.vadd
-@[measurability]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable])) const_nsmul]
+protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
+ AEStronglyMeasurable (f ^ n) μ :=
+ ⟨hf.mk f ^ n, hf.stronglyMeasurable_mk.pow _, hf.ae_eq_mk.pow_const _⟩
+
+@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
(hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.stronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AEStronglyMeasurable.const_smul
-@[measurability]
+@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
(hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ :=
hf.const_smul c
@@ -1695,7 +1695,7 @@ one can select a strongly measurable function as the almost everywhere limit. -/
theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
{f : ℕ → α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
- ∃ (f_lim : α → β) (hf_lim_meas : StronglyMeasurable f_lim),
+ ∃ f_lim : α → β, StronglyMeasurable f_lim ∧
∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) := by
borelize β
obtain ⟨g, _, hg⟩ :
∃ x ∈ s, _
instead of ∃ (x) (_ : x ∈ s), _
(#9184)
Search for [∀∃].*(_
and manually replace some occurrences with more readable versions.
In case of ∀
, the new expressions are defeq to the old ones.
In case of ∃
, they differ by exists_prop
.
In some rare cases, golf proofs that needed fixing.
@@ -907,12 +907,12 @@ theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β]
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hf_zero : ∀ x, x ∉ s → f x = 0) :
∃ fs : ℕ → α →ₛ β,
- (∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) ∧ ∀ (x) (_ : x ∉ s) (n), fs n x = 0 := by
+ (∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) ∧ ∀ x ∉ s, ∀ n, fs n x = 0 := by
let g_seq_s : ℕ → @SimpleFunc α m β := fun n => (hf.approx n).restrict s
have hg_eq : ∀ x ∈ s, ∀ n, g_seq_s n x = hf.approx n x := by
intro x hx n
rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_mem hx]
- have hg_zero : ∀ (x) (_ : x ∉ s), ∀ n, g_seq_s n x = 0 := by
+ have hg_zero : ∀ x ∉ s, ∀ n, g_seq_s n x = 0 := by
intro x hx n
rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_not_mem hx]
refine' ⟨g_seq_s, fun x => _, hg_zero⟩
@@ -929,7 +929,7 @@ on `s` is `m`-strongly-measurable, then `f` is also `m₂`-strongly-measurable.
theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableSpace α}
[TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
(hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
- (hf : StronglyMeasurable[m] f) (hf_zero : ∀ (x) (_ : x ∉ s), f x = 0) :
+ (hf : StronglyMeasurable[m] f) (hf_zero : ∀ x ∉ s, f x = 0) :
StronglyMeasurable[m₂] f := by
have hs_m₂ : MeasurableSet[m₂] s := by
rw [← Set.inter_univ s]
@@ -1699,7 +1699,7 @@ theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSp
∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) := by
borelize β
obtain ⟨g, _, hg⟩ :
- ∃ (g : α → β) (_ : Measurable g), ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) :=
+ ∃ g : α → β, Measurable g ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) :=
measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).aemeasurable) h_ae_tendsto
have Hg : AEStronglyMeasurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
refine' ⟨Hg.mk g, Hg.stronglyMeasurable_mk, _⟩
cases'
(#9171)
I literally went through and regex'd some uses of cases'
, replacing them with rcases
; this is meant to be a low effort PR as I hope that tools can do this in the future.
rcases
is an easier replacement than cases
, though with better tools we could in future do a second pass converting simple rcases
added here (and existing ones) to cases
.
@@ -244,7 +244,7 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
by_cases h0 : ‖hf.approx n x‖ = 0
· simp only [h0, _root_.div_zero, min_eq_right, zero_le_one, norm_zero, mul_zero]
exact hc
- cases' le_total ‖hf.approx n x‖ c with h h
+ rcases le_total ‖hf.approx n x‖ c with h | h
· rw [min_eq_left _]
· simpa only [norm_one, one_mul] using h
· rwa [one_le_div (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]
Finset.sup
, partialSups
etc (#8141)
Also rename Filter.Tendsto.sup_right_nhds
to Filter.Tendsto.sup_nhds
etc.
@@ -546,14 +546,14 @@ open Filter
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
- (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
@[aesop safe 20 (rule_sets [Measurable])]
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
- (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.inf MeasureTheory.StronglyMeasurable.inf
end Order
@@ -1120,7 +1120,7 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ := by
refine'
⟨fun n => hf.approx n ⊔ hg.approx n, fun n => _, fun x =>
- (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)⟩
refine' (measure_mono (support_sup _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
@@ -1130,7 +1130,7 @@ protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMe
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ := by
refine'
⟨fun n => hf.approx n ⊓ hg.approx n, fun n => _, fun x =>
- (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩
+ (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)⟩
refine' (measure_mono (support_inf _ _)).trans_lt _
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.inf
This is the supremum of
along with some minor fixes from failures on nightly-testing as Mathlib master
is merged into it.
Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.
I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0
branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.
In particular this includes adjustments for the Lean PRs
We can get rid of all the
local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)
macros across Mathlib (and in any projects that want to write natural number powers of reals).
Changes the default behaviour of simp
to (config := {decide := false})
. This makes simp
(and consequentially norm_num
) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp
or norm_num
to decide
or rfl
, or adding (config := {decide := true})
.
This changed the behaviour of simp
so that simp [f]
will only unfold "fully applied" occurrences of f
. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true })
. We may in future add a syntax for this, e.g. simp [!f]
; please provide feedback! In the meantime, we have made the following changes:
(config := { unfoldPartialApp := true })
in some places, to recover the old behaviour@[eqns]
to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp
and Function.flip
.This change in Lean may require further changes down the line (e.g. adding the !f
syntax, and/or upstreaming the special treatment for Function.comp
and Function.flip
, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!
Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>
@@ -987,7 +987,7 @@ theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measur
[TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ :=
⟨0, by
simp only [Pi.zero_apply, SimpleFunc.coe_zero, support_zero', measure_empty,
- WithTop.zero_lt_top, forall_const],
+ zero_lt_top, forall_const],
fun _ => tendsto_const_nhds⟩
#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zero
attribute [simp] ... in
-> attribute [local simp] ... in
(#7678)
Mathlib.Logic.Unique contains the line attribute [simp] eq_iff_true_of_subsingleton in ...
:
Despite what the in
part may imply, this adds the lemma to the simp set "globally", including for downstream files; it is likely that attribute [local simp] eq_iff_true_of_subsingleton in ...
was meant instead (or maybe scoped simp
, but I think "scoped" refers to the current namespace). Indeed, the relevant lemma is not marked with @[simp]
for possible slowness: https://github.com/leanprover/std4/blob/846e9e1d6bb534774d1acd2dc430e70987da3c18/Std/Logic.lean#L749. Adding it to the simp set causes the example at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Regression.20in.20simp to slow down.
This PR changes this and fixes the relevant downstream simp
s. There was also one ocurrence of attribute [simp] FullSubcategory.comp_def FullSubcategory.id_def in
in Mathlib.CategoryTheory.Monoidal.Subcategory but that was much easier to fix.
@@ -119,7 +119,7 @@ theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [Topologica
· exact ⟨fun _ => f_sf, fun x => tendsto_const_nhds⟩
· have h_univ : f ⁻¹' {x} = Set.univ := by
ext1 y
- simp
+ simp [eq_iff_true_of_subsingleton]
rw [h_univ]
exact MeasurableSet.univ
#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable
@@ -1542,6 +1542,11 @@ theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f
simp only [hfx, hgx]
#align measure_theory.ae_strongly_measurable.null_measurable_set_eq_fun MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
+@[to_additive]
+lemma nullMeasurableSet_mulSupport {E} [TopologicalSpace E] [MetrizableSpace E] [One E] {f : α → E}
+ (hf : AEStronglyMeasurable f μ) : NullMeasurableSet (mulSupport f) μ :=
+ (hf.nullMeasurableSet_eq_fun stronglyMeasurable_const.aestronglyMeasurable).compl
+
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a < g a } μ := by
If a measure μ
is sigma-finite, then μ.withDensity f
is sigma-finite for any a.e.-measurable and a.e. finite function f
.
@@ -971,29 +971,11 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
∃ s : ℕ → Set α,
(∀ n, MeasurableSet[m] (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ n) ∧
⋃ i, s i = Set.univ := by
- let sigma_finite_sets := spanningSets (μ.trim hm)
- let norm_sets := fun n : ℕ => { x | ‖f x‖ ≤ n }
- have norm_sets_spanning : ⋃ n, norm_sets n = Set.univ := by
- ext1 x
- simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
- exact ⟨⌈‖f x‖⌉₊, Nat.le_ceil ‖f x‖⟩
- let sets n := sigma_finite_sets n ∩ norm_sets n
- have h_meas : ∀ n, MeasurableSet[m] (sets n) := by
- refine' fun n => MeasurableSet.inter _ _
- · exact measurable_spanningSets (μ.trim hm) n
- · exact hf.norm.measurableSet_le stronglyMeasurable_const
- have h_finite : ∀ n, μ (sets n) < ∞ := by
- refine' fun n => (measure_mono (Set.inter_subset_left _ _)).trans_lt _
- exact (le_trim hm).trans_lt (measure_spanningSets_lt_top (μ.trim hm) n)
- refine' ⟨sets, fun n => ⟨h_meas n, h_finite n, _⟩, _⟩
- · exact fun x hx => hx.2
- · have :
- ⋃ i, sigma_finite_sets i ∩ norm_sets i = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i := by
- refine' Set.iUnion_inter_of_monotone (monotone_spanningSets (μ.trim hm)) fun i j hij x => _
- simp only [Set.mem_setOf_eq]
- refine' fun hif => hif.trans _
- exact_mod_cast hij
- rw [this, norm_sets_spanning, iUnion_spanningSets (μ.trim hm), Set.inter_univ]
+ obtain ⟨s, hs, hs_univ⟩ := exists_spanning_measurableSet_le hf.nnnorm.measurable (μ.trim hm)
+ refine ⟨s, fun n ↦ ⟨(hs n).1, (le_trim hm).trans_lt (hs n).2.1, fun x hx ↦ ?_⟩, hs_univ⟩
+ have hx_nnnorm : ‖f x‖₊ ≤ n := (hs n).2.2 x hx
+ rw [← coe_nnnorm]
+ norm_cast
#align measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le
end StronglyMeasurable
@@ -1806,8 +1806,7 @@ theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
AEStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
AEStronglyMeasurable f (μ.restrict <| Ioc b a) :=
by rw [uIoc_eq_union, aestronglyMeasurable_union_iff]
-#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff
-MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
+#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
@[measurability]
theorem smul_measure {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
@@ -1879,8 +1878,7 @@ theorem _root_.ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => L (f x) (g x)) μ :=
L.continuous₂.comp_aestronglyMeasurable <| hf.prod_mk hg
-#align continuous_linear_map.ae_strongly_measurable_comp₂
-ContinuousLinearMap.aestronglyMeasurable_comp₂
+#align continuous_linear_map.ae_strongly_measurable_comp₂ ContinuousLinearMap.aestronglyMeasurable_comp₂
end ContinuousLinearMapNontriviallyNormedField
@@ -1936,16 +1934,14 @@ theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk
-MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
@[aesop 10% apply (rule_sets [Measurable])]
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
AEMeasurable f μ :=
⟨hf.mk f, hf.finStronglyMeasurable_mk.measurable, hf.ae_eq_mk⟩
-#align measure_theory.ae_fin_strongly_measurable.ae_measurable
-MeasureTheory.AEFinStronglyMeasurable.aemeasurable
+#align measure_theory.ae_fin_strongly_measurable.ae_measurable MeasureTheory.AEFinStronglyMeasurable.aemeasurable
end Mk
@@ -1720,6 +1720,27 @@ theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSp
rwa [h'x] at hx
#align exists_strongly_measurable_limit_of_tendsto_ae exists_stronglyMeasurable_limit_of_tendsto_ae
+theorem piecewise {s : Set α} [DecidablePred (· ∈ s)]
+ (hs : MeasurableSet s) (hf : AEStronglyMeasurable f (μ.restrict s))
+ (hg : AEStronglyMeasurable g (μ.restrict sᶜ)) :
+ AEStronglyMeasurable (s.piecewise f g) μ := by
+ refine ⟨s.piecewise (hf.mk f) (hg.mk g),
+ StronglyMeasurable.piecewise hs hf.stronglyMeasurable_mk hg.stronglyMeasurable_mk, ?_⟩
+ refine ae_of_ae_restrict_of_ae_restrict_compl s ?_ ?_
+ · have h := hf.ae_eq_mk
+ rw [Filter.EventuallyEq, ae_restrict_iff' hs] at h
+ rw [ae_restrict_iff' hs]
+ filter_upwards [h] with x hx
+ intro hx_mem
+ simp only [hx_mem, Set.piecewise_eq_of_mem, hx hx_mem]
+ · have h := hg.ae_eq_mk
+ rw [Filter.EventuallyEq, ae_restrict_iff' hs.compl] at h
+ rw [ae_restrict_iff' hs.compl]
+ filter_upwards [h] with x hx
+ intro hx_mem
+ rw [Set.mem_compl_iff] at hx_mem
+ simp only [hx_mem, not_false_eq_true, Set.piecewise_eq_of_not_mem, hx hx_mem]
+
theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
(h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) := by
borelize β
@@ -5,7 +5,6 @@ Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
-import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFuncDense
@@ -699,14 +698,38 @@ theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSp
· exact hf.measurable.stronglyMeasurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
-/-- A continuous function with compact support is strongly measurable. -/
-theorem _root_.Continuous.stronglyMeasurable_of_hasCompactSupport
+/-- A continuous function whose support is contained in a compact set is strongly measurable. -/
+@[to_additive]
+theorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
[MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
- [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [Zero β] {f : α → β}
- (hf : Continuous f) (h'f : HasCompactSupport f) : StronglyMeasurable f := by
+ [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
+ (hf : Continuous f) {k : Set α} (hk : IsCompact k)
+ (h'f : mulSupport f ⊆ k) : StronglyMeasurable f := by
letI : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
rw [stronglyMeasurable_iff_measurable_separable]
- exact ⟨hf.measurable, IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)⟩
+ exact ⟨hf.measurable, (isCompact_range_of_mulSupport_subset_isCompact hf hk h'f).isSeparable⟩
+
+/-- A continuous function with compact support is strongly measurable. -/
+@[to_additive]
+theorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport
+ [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
+ [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
+ (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f :=
+ hf.stronglyMeasurable_of_mulSupport_subset_isCompact h'f (subset_mulTSupport f)
+
+/-- A continuous function with compact support on a product space is strongly measurable for the
+product sigma-algebra. The subtlety is that we do not assume that the spaces are separable, so the
+product of the Borel sigma algebras might not contain all open sets, but still it contains enough
+of them to approximate compactly supported continuous functions. -/
+lemma _root_.HasCompactSupport.stronglyMeasurable_of_prod {X Y : Type*} [Zero α]
+ [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y]
+ [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace α] [PseudoMetrizableSpace α]
+ {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f) :
+ StronglyMeasurable f := by
+ borelize α
+ apply stronglyMeasurable_iff_measurable_separable.2 ⟨h'f.measurable_of_prod hf, ?_⟩
+ letI : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
+ exact IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
@@ -706,7 +706,7 @@ theorem _root_.Continuous.stronglyMeasurable_of_hasCompactSupport
(hf : Continuous f) (h'f : HasCompactSupport f) : StronglyMeasurable f := by
letI : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
rw [stronglyMeasurable_iff_measurable_separable]
- refine ⟨hf.measurable, IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)⟩
+ exact ⟨hf.measurable, IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)⟩
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
rcases
, convert
and congrm
(#7725)
Replace rcases(
with rcases (
. Same thing for convert(
and congrm(
. No other change.
@@ -1648,7 +1648,7 @@ theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β
exact mem_range_self x }
have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.aemeasurable
exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this
- · rcases(aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
+ · rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
exact ⟨g ⁻¹' t, hg.isSeparable_preimage ht, h't⟩
#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iff
filter_upwards
(#7719)
mathport was forgetting a space in filter_upwards [...]with
instead of filter_upwards [...] with
.
@@ -1709,7 +1709,7 @@ theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι
refine' ⟨⋃ i, t i, isSeparable_iUnion t_sep, _⟩
simp only [Measure.ae_sum_eq, mem_iUnion, eventually_iSup]
intro i
- filter_upwards [ht i]with x hx
+ filter_upwards [ht i] with x hx
exact ⟨i, hx⟩
#align measure_theory.ae_strongly_measurable.sum_measure MeasureTheory.AEStronglyMeasurable.sum_measure
NullMeasurable
function is AEMeasurable
(#7604)
This is true if the function admits an a.e. range with countably generated σ-algebra.
In particular, a function is AEStronglyMeasurable
iff it is NullMeasurable
and it admits a separable a.e. range.
@@ -1612,6 +1612,14 @@ theorem _root_.aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizable
exact stronglyMeasurable_iff_measurable_separable.2 ⟨g_meas, t_sep.mono gt⟩
#align ae_strongly_measurable_iff_ae_measurable_separable aestronglyMeasurable_iff_aemeasurable_separable
+theorem _root_.aestronglyMeasurable_iff_nullMeasurable_separable [PseudoMetrizableSpace β]
+ [MeasurableSpace β] [BorelSpace β] :
+ AEStronglyMeasurable f μ ↔
+ NullMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t :=
+ aestronglyMeasurable_iff_aemeasurable_separable.trans <| and_congr_left fun ⟨_, hsep, h⟩ ↦
+ have := hsep.secondCountableTopology
+ ⟨AEMeasurable.nullMeasurable, fun hf ↦ hf.aemeasurable_of_aerange h⟩
+
theorem _root_.MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type*}
{mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ}
(hf : MeasurableEmbedding f) {g : α → β} :
@@ -654,12 +654,9 @@ theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetriz
[SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
StronglyMeasurable f := by
letI := pseudoMetrizableSpacePseudoMetric β
- rcases isEmpty_or_nonempty β with ⟨⟩ <;> skip
- · exact Subsingleton.stronglyMeasurable f
- · inhabit β
- exact
- ⟨SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x =>
- SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by rw [closure_univ]; simp)⟩
+ nontriviality β; inhabit β
+ exact ⟨SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x ↦
+ SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by rw [closure_univ]; simp)⟩
#align measurable.strongly_measurable Measurable.stronglyMeasurable
/-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
@@ -681,26 +678,10 @@ range. -/
theorem _root_.stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α}
[TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] :
StronglyMeasurable f ↔ Measurable f ∧ IsSeparable (range f) := by
- refine' ⟨fun H => ⟨H.measurable, H.isSeparable_range⟩, _⟩
- rintro ⟨H, H'⟩
- letI := pseudoMetrizableSpacePseudoMetric β
- let g := codRestrict f (closure (range f)) fun x => subset_closure (mem_range_self x)
- have fg : f = ((↑) : closure (range f) → β) ∘ g := by
- ext x
- rfl
- have T : MeasurableEmbedding ((↑) : closure (range f) → β) := by
- apply ClosedEmbedding.measurableEmbedding
- exact closedEmbedding_subtype_val isClosed_closure
- have g_meas : Measurable g := by
- rw [fg] at H
- exact T.measurable_comp_iff.1 H
- have : SecondCountableTopology (closure (range f)) := by
- suffices SeparableSpace (closure (range f)) by
- exact UniformSpace.secondCountable_of_separable _
- exact (IsSeparable.closure H').separableSpace
- have g_smeas : StronglyMeasurable g := Measurable.stronglyMeasurable g_meas
- rw [fg]
- exact continuous_subtype_val.comp_stronglyMeasurable g_smeas
+ refine ⟨fun H ↦ ⟨H.measurable, H.isSeparable_range⟩, fun ⟨Hm, Hsep⟩ ↦ ?_⟩
+ have := Hsep.secondCountableTopology
+ have Hm' : StronglyMeasurable (rangeFactorization f) := Hm.subtype_mk.stronglyMeasurable
+ exact continuous_subtype_val.comp_stronglyMeasurable Hm'
#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separable
/-- A continuous function is strongly measurable when either the source space or the target space
@@ -736,20 +717,16 @@ theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [T
refine'
⟨fun H => stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩, fun H =>
hg.continuous.comp_stronglyMeasurable H⟩
- · let G : β → range g := codRestrict g (range g) mem_range_self
+ · let G : β → range g := rangeFactorization g
have hG : ClosedEmbedding G :=
{ hg.codRestrict _ _ with
closed_range := by
- convert isClosed_univ (α := ↥(range g))
- apply eq_univ_of_forall
- rintro ⟨-, ⟨x, rfl⟩⟩
- exact mem_range_self x }
+ rw [surjective_onto_range.range_eq]
+ exact isClosed_univ }
have : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
exact hG.measurableEmbedding.measurable_comp_iff.1 this
· have : IsSeparable (g ⁻¹' range (g ∘ f)) := hg.isSeparable_preimage H.isSeparable_range
- convert this
- ext x
- simp [hg.inj.eq_iff]
+ rwa [range_comp, hg.inj.preimage_image] at this
#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff
/-- A sequential limit of strongly measurable functions is strongly measurable. -/
@@ -794,49 +771,46 @@ protected theorem ite {_ : MeasurableSpace α} [TopologicalSpace β] {p : α →
StronglyMeasurable.piecewise hp hf hg
#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite
+@[measurability]
+theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
+ {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
+ (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
+ StronglyMeasurable (Function.extend g f g') := by
+ refine' ⟨fun n => SimpleFunc.extend (hf.approx n) g hg (hg'.approx n), _⟩
+ intro x
+ by_cases hx : ∃ y, g y = x
+ · rcases hx with ⟨y, rfl⟩
+ simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using
+ hf.tendsto_approx y
+ · simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using
+ hg'.tendsto_approx x
+#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
+
+theorem _root_.MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
+ {_ : MeasurableSpace α} {_ : MeasurableSpace γ} [TopologicalSpace β]
+ (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
+ ∃ f' : γ → β, StronglyMeasurable f' ∧ f' ∘ g = f :=
+ ⟨Function.extend g f fun x => Classical.choice (hne x),
+ hg.stronglyMeasurable_extend hf (stronglyMeasurable_const' fun _ _ => rfl),
+ funext fun _ => hg.injective.extend_apply _ _ _⟩
+#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend
+
theorem _root_.stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
(h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
(hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by
- classical
- let f : ℕ → α →ₛ β := fun n =>
- { toFun := fun x =>
- if hx : x ∈ s then hc.approx n ⟨x, hx⟩
- else hd.approx n ⟨x, by simpa [hx] using h (mem_univ x)⟩
- measurableSet_fiber' := by
- intro x
- convert(hs.subtype_image ((hc.approx n).measurableSet_fiber x)).union
- ((ht.subtype_image ((hd.approx n).measurableSet_fiber x)).diff hs)
- ext1 y
- simp only [mem_union, mem_preimage, mem_singleton_iff, mem_image, SetCoe.exists,
- Subtype.coe_mk, exists_and_right, exists_eq_right, mem_diff]
- by_cases hy : y ∈ s
- · rw [dif_pos hy]
- simp only [hy, exists_true_left, not_true, and_false_iff, or_false_iff]
- · rw [dif_neg hy]
- have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- simp only [A, hy, false_or_iff, IsEmpty.exists_iff, not_false_iff, and_true_iff,
- exists_true_left]
- finite_range' := by
- apply ((hc.approx n).finite_range.union (hd.approx n).finite_range).subset
- rintro - ⟨y, rfl⟩
- dsimp
- by_cases hy : y ∈ s
- · left
- rw [dif_pos hy]
- exact mem_range_self _
- · right
- rw [dif_neg hy]
- exact mem_range_self _ }
- refine' ⟨f, fun y => _⟩
- by_cases hy : y ∈ s
- · convert hc.tendsto_approx ⟨y, hy⟩ using 1
- ext1 n
- simp only [dif_pos hy, SimpleFunc.apply_mk]
- · have A : y ∈ t := by simpa [hy] using h (mem_univ y)
- convert hd.tendsto_approx ⟨y, A⟩ using 1
- ext1 n
- simp only [dif_neg hy, SimpleFunc.apply_mk]
+ nontriviality β; inhabit β
+ suffices Function.extend Subtype.val (fun x : s ↦ f x)
+ (Function.extend (↑) (fun x : t ↦ f x) fun _ ↦ default) = f from
+ this ▸ (MeasurableEmbedding.subtype_coe hs).stronglyMeasurable_extend hc <|
+ (MeasurableEmbedding.subtype_coe ht).stronglyMeasurable_extend hd stronglyMeasurable_const
+ ext x
+ by_cases hxs : x ∈ s
+ · lift x to s using hxs
+ simp [Subtype.coe_injective.extend_apply]
+ · lift x to t using (h trivial).resolve_left hxs
+ rw [extend_apply', Subtype.coe_injective.extend_apply]
+ exact fun ⟨y, hy⟩ ↦ hxs <| hy ▸ y.2
#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover
theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace α}
@@ -885,30 +859,6 @@ protected theorem real_toNNReal {_ : MeasurableSpace α} {f : α → ℝ} (hf :
continuous_real_toNNReal.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal
-@[measurability]
-theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
- {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
- (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
- StronglyMeasurable (Function.extend g f g') := by
- refine' ⟨fun n => SimpleFunc.extend (hf.approx n) g hg (hg'.approx n), _⟩
- intro x
- by_cases hx : ∃ y, g y = x
- · rcases hx with ⟨y, rfl⟩
- simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using
- hf.tendsto_approx y
- · simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using
- hg'.tendsto_approx x
-#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
-
-theorem _root_.MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
- {_ : MeasurableSpace α} {_ : MeasurableSpace γ} [TopologicalSpace β]
- (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
- ∃ f' : γ → β, StronglyMeasurable f' ∧ f' ∘ g = f :=
- ⟨Function.extend g f fun x => Classical.choice (hne x),
- hg.stronglyMeasurable_extend hf (stronglyMeasurable_const' fun _ _ => rfl),
- funext fun _ => hg.injective.extend_apply _ _ _⟩
-#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend
-
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
{f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
MeasurableSet { x | f x = g x } := by
It was in Integral.Lebesgue, which is about the definition of the Lebesgue integral.
@@ -6,7 +6,7 @@ Authors: Rémy Degenne, Sébastien Gouëzel
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
-import Mathlib.MeasureTheory.Integral.Lebesgue
+import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
We already have the strong law of large numbers for real-valued integrable random variables. We generalize it to general vector-valued integrable random variables. This does not require any second-countability assumptions as integrable functions can by definition be approximated by simple functions, for which the result is deduced from the one-dimensional one.
Along the way, we extend a few lemmas in the library from the real case to the vector case, and remove unneeded second-countability assumptions.
@@ -467,6 +467,45 @@ protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Cont
#align measure_theory.strongly_measurable.smul_const MeasureTheory.StronglyMeasurable.smul_const
#align measure_theory.strongly_measurable.vadd_const MeasureTheory.StronglyMeasurable.vadd_const
+/-- In a normed vector space, the addition of a measurable function and a strongly measurable
+function is measurable. Note that this is not true without further second-countability assumptions
+for the addition of two measurable functions. -/
+theorem _root_.Measurable.add_stronglyMeasurable
+ {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
+ [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
+ {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
+ Measurable (g + f) := by
+ rcases hf with ⟨φ, hφ⟩
+ have : Tendsto (fun n x ↦ g x + φ n x) atTop (𝓝 (g + f)) :=
+ tendsto_pi_nhds.2 (fun x ↦ tendsto_const_nhds.add (hφ x))
+ apply measurable_of_tendsto_metrizable (fun n ↦ ?_) this
+ exact hg.add_simpleFunc _
+
+/-- In a normed vector space, the subtraction of a measurable function and a strongly measurable
+function is measurable. Note that this is not true without further second-countability assumptions
+for the subtraction of two measurable functions. -/
+theorem _root_.Measurable.sub_stronglyMeasurable
+ {α E : Type*} {_ : MeasurableSpace α} [AddCommGroup E] [TopologicalSpace E]
+ [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [ContinuousNeg E] [PseudoMetrizableSpace E]
+ {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
+ Measurable (g - f) := by
+ rw [sub_eq_add_neg]
+ exact hg.add_stronglyMeasurable hf.neg
+
+/-- In a normed vector space, the addition of a strongly measurable function and a measurable
+function is measurable. Note that this is not true without further second-countability assumptions
+for the addition of two measurable functions. -/
+theorem _root_.Measurable.stronglyMeasurable_add
+ {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
+ [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
+ {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
+ Measurable (f + g) := by
+ rcases hf with ⟨φ, hφ⟩
+ have : Tendsto (fun n x ↦ φ n x + g x) atTop (𝓝 (f + g)) :=
+ tendsto_pi_nhds.2 (fun x ↦ (hφ x).add tendsto_const_nhds)
+ apply measurable_of_tendsto_metrizable (fun n ↦ ?_) this
+ exact hg.simpleFunc_add _
+
end Arithmetic
section MulAction
The main result is that the integral (i.e. the expected value) with regard to a
measure derived from a Pmf
is a sum weighted by the Pmf
.
It also provides the expected value for specific probability mass functions (bernoulli so far).
@@ -135,6 +135,11 @@ theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [Top
⟨fun _ => SimpleFunc.ofIsEmpty, isEmptyElim⟩
#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.stronglyMeasurable_of_isEmpty
+theorem stronglyMeasurable_of_fintype [Fintype α] {_ : MeasurableSpace α}
+ [MeasurableSingletonClass α] [TopologicalSpace β]
+ (f : α → β) : StronglyMeasurable f :=
+ ⟨fun _ => SimpleFunc.ofFintype f, fun _ => tendsto_const_nhds⟩
+
theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSpace β] {b : β} :
StronglyMeasurable fun _ : α => b :=
⟨fun _ => SimpleFunc.const α b, fun _ => tendsto_const_nhds⟩
MeasureTheory.snorm_sub_le'
.neg
lemmas up so that we can use them in MeasureTheory.snorm_sub_le'
.@@ -422,7 +422,7 @@ theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c :
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
@[to_additive (attr := measurability)]
-protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
+protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
⟨fun n => (hf.approx n)⁻¹, fun x => (hf.tendsto_approx x).inv⟩
#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv
@@ -1319,7 +1319,7 @@ protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurab
#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AEStronglyMeasurable.const_add
@[to_additive (attr := measurability)]
-protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
+protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable f⁻¹ μ :=
⟨(hf.mk f)⁻¹, hf.stronglyMeasurable_mk.inv, hf.ae_eq_mk.inv⟩
#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
We have currently that the product of two Borel spaces is Borel when both of them are second-countable. It is in fact sufficient to assume that only one of them is second-countable. We prove this in this PR.
Also move the definition of SecondCountableEither
from Function.StronglyMeasurable
to BorelSpace.Basic
to be able to use it in the statement of the above theorem.
@@ -64,25 +64,6 @@ open MeasureTheory Filter TopologicalSpace Function Set MeasureTheory.Measure
open ENNReal Topology MeasureTheory NNReal BigOperators
-/-- The typeclass `SecondCountableTopologyEither α β` registers the fact that at least one of
-the two spaces has second countable topology. This is the right assumption to ensure that continuous
-maps from `α` to `β` are strongly measurable. -/
-class SecondCountableTopologyEither (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] :
- Prop where
- /-- The projection out of `SecondCountableTopologyEither` -/
- out : SecondCountableTopology α ∨ SecondCountableTopology β
-#align second_countable_topology_either SecondCountableTopologyEither
-
-instance (priority := 100) secondCountableTopologyEither_of_left (α β : Type*) [TopologicalSpace α]
- [TopologicalSpace β] [SecondCountableTopology α] : SecondCountableTopologyEither α β
- where out := Or.inl (by infer_instance)
-#align second_countable_topology_either_of_left secondCountableTopologyEither_of_left
-
-instance (priority := 100) secondCountableTopologyEither_of_right (α β : Type*)
- [TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] :
- SecondCountableTopologyEither α β where out := Or.inr (by infer_instance)
-#align second_countable_topology_either_of_right secondCountableTopologyEither_of_right
-
variable {α β γ ι : Type*} [Countable ι]
namespace MeasureTheory
MulZeroClass.
in mul_zero
/zero_mul
(#6682)
Search&replace MulZeroClass.mul_zero
-> mul_zero
, MulZeroClass.zero_mul
-> zero_mul
.
These were introduced by Mathport, as the full name of mul_zero
is actually MulZeroClass.mul_zero
(it's exported with the short name).
@@ -257,7 +257,7 @@ theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [
simp only [StronglyMeasurable.approxBounded, SimpleFunc.coe_map, Function.comp_apply]
refine' (norm_smul_le _ _).trans _
by_cases h0 : ‖hf.approx n x‖ = 0
- · simp only [h0, _root_.div_zero, min_eq_right, zero_le_one, norm_zero, MulZeroClass.mul_zero]
+ · simp only [h0, _root_.div_zero, min_eq_right, zero_le_one, norm_zero, mul_zero]
exact hc
cases' le_total ‖hf.approx n x‖ c with h h
· rw [min_eq_left _]
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -67,23 +67,23 @@ open ENNReal Topology MeasureTheory NNReal BigOperators
/-- The typeclass `SecondCountableTopologyEither α β` registers the fact that at least one of
the two spaces has second countable topology. This is the right assumption to ensure that continuous
maps from `α` to `β` are strongly measurable. -/
-class SecondCountableTopologyEither (α β : Type _) [TopologicalSpace α] [TopologicalSpace β] :
+class SecondCountableTopologyEither (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] :
Prop where
/-- The projection out of `SecondCountableTopologyEither` -/
out : SecondCountableTopology α ∨ SecondCountableTopology β
#align second_countable_topology_either SecondCountableTopologyEither
-instance (priority := 100) secondCountableTopologyEither_of_left (α β : Type _) [TopologicalSpace α]
+instance (priority := 100) secondCountableTopologyEither_of_left (α β : Type*) [TopologicalSpace α]
[TopologicalSpace β] [SecondCountableTopology α] : SecondCountableTopologyEither α β
where out := Or.inl (by infer_instance)
#align second_countable_topology_either_of_left secondCountableTopologyEither_of_left
-instance (priority := 100) secondCountableTopologyEither_of_right (α β : Type _)
+instance (priority := 100) secondCountableTopologyEither_of_right (α β : Type*)
[TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] :
SecondCountableTopologyEither α β where out := Or.inr (by infer_instance)
#align second_countable_topology_either_of_right secondCountableTopologyEither_of_right
-variable {α β γ ι : Type _} [Countable ι]
+variable {α β γ ι : Type*} [Countable ι]
namespace MeasureTheory
@@ -485,7 +485,7 @@ end Arithmetic
section MulAction
-variable {M G G₀ : Type _}
+variable {M G G₀ : Type*}
variable [TopologicalSpace β]
variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
@@ -541,7 +541,7 @@ end Order
section Monoid
-variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
+variable {M : Type*} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
@@ -565,7 +565,7 @@ end Monoid
section CommMonoid
-variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
+variable {M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
@@ -584,14 +584,14 @@ theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum
@[to_additive (attr := measurability)]
-theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i in s, f i) :=
Finset.prod_induction _ _ (fun _a _b ha hb => ha.mul hb) (@stronglyMeasurable_one α M _ _ _) hf
#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'
#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'
@[to_additive (attr := measurability)]
-theorem _root_.Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i in s, f i a := by
simpa only [← Finset.prod_apply] using s.stronglyMeasurable_prod' hf
#align finset.strongly_measurable_prod Finset.stronglyMeasurable_prod
@@ -728,7 +728,7 @@ theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [T
#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff
/-- A sequential limit of strongly measurable functions is strongly measurable. -/
-theorem _root_.stronglyMeasurable_of_tendsto {ι : Type _} {m : MeasurableSpace α}
+theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace α}
[TopologicalSpace β] [PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
{f : ι → α → β} {g : α → β} (hf : ∀ i, StronglyMeasurable (f i)) (lim : Tendsto f u (𝓝 g)) :
StronglyMeasurable g := by
@@ -830,26 +830,26 @@ protected theorem indicator {_ : MeasurableSpace α} [TopologicalSpace β] [Zero
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
@[aesop safe 20 apply (rule_sets [Measurable])]
-protected theorem dist {_ : MeasurableSpace α} {β : Type _} [PseudoMetricSpace β] {f g : α → β}
+protected theorem dist {_ : MeasurableSpace α} {β : Type*} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist
@[measurability]
-protected theorem norm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
+protected theorem norm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ :=
continuous_norm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm
@[measurability]
-protected theorem nnnorm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
+protected theorem nnnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ :=
continuous_nnnorm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm
@[measurability]
-protected theorem ennnorm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
+protected theorem ennnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
(ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
@@ -1406,7 +1406,7 @@ end Order
section Monoid
-variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
+variable {M : Type*} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
@@ -1430,7 +1430,7 @@ end Monoid
section CommMonoid
-variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
+variable {M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
@@ -1449,7 +1449,7 @@ theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.ae_strongly_measurable_sum Multiset.aestronglyMeasurable_sum
@[to_additive (attr := measurability)]
-theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i in s, f i) μ :=
Multiset.aestronglyMeasurable_prod' _ fun _g hg =>
let ⟨_i, hi, hg⟩ := Multiset.mem_map.1 hg
@@ -1458,7 +1458,7 @@ theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α
#align finset.ae_strongly_measurable_sum' Finset.aestronglyMeasurable_sum'
@[to_additive (attr := measurability)]
-theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
+theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
AEStronglyMeasurable (fun a => ∏ i in s, f i a) μ := by
simpa only [← Finset.prod_apply] using s.aestronglyMeasurable_prod' hf
@@ -1479,7 +1479,7 @@ theorem _root_.AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [Ope
#align ae_measurable.ae_strongly_measurable AEMeasurable.aestronglyMeasurable
@[measurability]
-theorem _root_.aestronglyMeasurable_id {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
+theorem _root_.aestronglyMeasurable_id {α : Type*} [TopologicalSpace α] [PseudoMetrizableSpace α]
{_ : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
AEStronglyMeasurable (id : α → α) μ :=
aemeasurable_id.aestronglyMeasurable
@@ -1494,32 +1494,32 @@ theorem _root_.aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β]
end SecondCountableAEStronglyMeasurable
@[aesop safe 20 apply (rule_sets [Measurable])]
-protected theorem dist {β : Type _} [PseudoMetricSpace β] {f g : α → β}
+protected theorem dist {β : Type*} [PseudoMetricSpace β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
continuous_dist.comp_aestronglyMeasurable (hf.prod_mk hg)
#align measure_theory.ae_strongly_measurable.dist MeasureTheory.AEStronglyMeasurable.dist
@[measurability]
-protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
+protected theorem norm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖) μ :=
continuous_norm.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AEStronglyMeasurable.norm
@[measurability]
-protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
+protected theorem nnnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ :=
continuous_nnnorm.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnorm
@[measurability]
-protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
+protected theorem ennnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
(ENNReal.continuous_coe.comp_aestronglyMeasurable hf.nnnorm).aemeasurable
#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AEStronglyMeasurable.ennnorm
@[aesop safe 20 apply (rule_sets [Measurable])]
-protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α → β}
+protected theorem edist {β : Type*} [SeminormedAddCommGroup β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEMeasurable (fun a => edist (f a) (g a)) μ :=
(continuous_edist.comp_aestronglyMeasurable (hf.prod_mk hg)).aemeasurable
@@ -1588,26 +1588,26 @@ theorem _root_.aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : Me
⟨hf.mk f, StronglyMeasurable.mono hf.stronglyMeasurable_mk hm, ae_eq_of_ae_eq_trim hf.ae_eq_mk⟩
#align ae_strongly_measurable_of_ae_strongly_measurable_trim aestronglyMeasurable_of_aestronglyMeasurable_trim
-theorem comp_aemeasurable {γ : Type _} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
+theorem comp_aemeasurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
AEStronglyMeasurable (g ∘ f) μ :=
⟨hg.mk g ∘ hf.mk f, hg.stronglyMeasurable_mk.comp_measurable hf.measurable_mk,
(ae_eq_comp hf hg.ae_eq_mk).trans (hf.ae_eq_mk.fun_comp (hg.mk g))⟩
#align measure_theory.ae_strongly_measurable.comp_ae_measurable MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
-theorem comp_measurable {γ : Type _} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
+theorem comp_measurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
{μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
AEStronglyMeasurable (g ∘ f) μ :=
hg.comp_aemeasurable hf.aemeasurable
#align measure_theory.ae_strongly_measurable.comp_measurable MeasureTheory.AEStronglyMeasurable.comp_measurable
-theorem comp_quasiMeasurePreserving {γ : Type _} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
+theorem comp_quasiMeasurePreserving {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
{f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
(hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
(hg.mono' hf.absolutelyContinuous).comp_measurable hf.measurable
#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
-theorem comp_measurePreserving {γ : Type _} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
+theorem comp_measurePreserving {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
{f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
(hf : MeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
hg.comp_quasiMeasurePreserving hf.quasiMeasurePreserving
@@ -1637,7 +1637,7 @@ theorem _root_.aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizable
exact stronglyMeasurable_iff_measurable_separable.2 ⟨g_meas, t_sep.mono gt⟩
#align ae_strongly_measurable_iff_ae_measurable_separable aestronglyMeasurable_iff_aemeasurable_separable
-theorem _root_.MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type _}
+theorem _root_.MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type*}
{mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ}
(hf : MeasurableEmbedding f) {g : α → β} :
AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ := by
@@ -1669,7 +1669,7 @@ theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β
exact ⟨g ⁻¹' t, hg.isSeparable_preimage ht, h't⟩
#align embedding.ae_strongly_measurable_comp_iff Embedding.aestronglyMeasurable_comp_iff
-theorem _root_.MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type _}
+theorem _root_.MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type*}
{f : α → β} {mα : MeasurableSpace α} {μa : Measure α} {mβ : MeasurableSpace β} {μb : Measure β}
(hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ} :
AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb := by
@@ -1678,7 +1678,7 @@ theorem _root_.MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β
/-- An almost everywhere sequential limit of almost everywhere strongly measurable functions is
almost everywhere strongly measurable. -/
-theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type _} [PseudoMetrizableSpace β]
+theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizableSpace β]
(u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
(hf : ∀ i, AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
AEStronglyMeasurable g μ := by
@@ -1783,16 +1783,16 @@ theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
@[measurability]
-theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
+theorem smul_measure {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
#align measure_theory.ae_strongly_measurable.smul_measure MeasureTheory.AEStronglyMeasurable.smul_measure
section NormedSpace
-variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
theorem _root_.aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ :=
@@ -1803,7 +1803,7 @@ end NormedSpace
section MulAction
-variable {M G G₀ : Type _}
+variable {M G G₀ : Type*}
variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
@@ -1828,13 +1828,13 @@ end MulAction
section ContinuousLinearMapNontriviallyNormedField
-variable {𝕜 : Type _} [NontriviallyNormedField 𝕜]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
theorem _root_.StronglyMeasurable.apply_continuousLinearMap {_m : MeasurableSpace α}
{φ : α → F →L[𝕜] E}
@@ -1857,7 +1857,7 @@ ContinuousLinearMap.aestronglyMeasurable_comp₂
end ContinuousLinearMapNontriviallyNormedField
-theorem _root_.aestronglyMeasurable_withDensity_iff {E : Type _} [NormedAddCommGroup E]
+theorem _root_.aestronglyMeasurable_withDensity_iff {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEStronglyMeasurable (fun x => (f x : ℝ) • g x) μ := by
@@ -2016,7 +2016,7 @@ end AEFinStronglyMeasurable
section SecondCountableTopology
-variable {G : Type _} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : Measure α}
+variable {G : Type*} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : Measure α}
[SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G]
{f : α → G}
@@ -2050,7 +2050,7 @@ theorem aefinStronglyMeasurable_of_aemeasurable {_m0 : MeasurableSpace α} (μ :
end SecondCountableTopology
-theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [TopologicalSpace ι]
+theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type*} [TopologicalSpace ι]
[MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
{mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
{m : MeasurableSpace α} {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x)
@@ -2082,7 +2082,7 @@ theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [Topo
exact ((t_sf n).measurable.comp measurable_fst).subtype_mk
#align measure_theory.measurable_uncurry_of_continuous_of_measurable MeasureTheory.measurable_uncurry_of_continuous_of_measurable
-theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type _}
+theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
[TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
[OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
{u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
@@ -681,7 +681,7 @@ theorem _root_.stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace
/-- A continuous function is strongly measurable when either the source space or the target space
is second-countable. -/
theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
- [OpensMeasurableSpace α] {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
+ [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
[h : SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f) :
StronglyMeasurable f := by
borelize β
@@ -693,6 +693,15 @@ theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSp
· exact hf.measurable.stronglyMeasurable
#align continuous.strongly_measurable Continuous.stronglyMeasurable
+/-- A continuous function with compact support is strongly measurable. -/
+theorem _root_.Continuous.stronglyMeasurable_of_hasCompactSupport
+ [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
+ [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [Zero β] {f : α → β}
+ (hf : Continuous f) (h'f : HasCompactSupport f) : StronglyMeasurable f := by
+ letI : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
+ rw [stronglyMeasurable_iff_measurable_separable]
+ refine ⟨hf.measurable, IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)⟩
+
/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
@@ -2,11 +2,6 @@
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit ef95945cd48c932c9e034872bd25c3c220d9c946
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
@@ -14,6 +9,8 @@ import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Function.SimpleFuncDense
+#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"ef95945cd48c932c9e034872bd25c3c220d9c946"
+
/-!
# Strongly measurable and finitely strongly measurable functions
@@ -1601,6 +1601,11 @@ theorem comp_quasiMeasurePreserving {γ : Type _} {_ : MeasurableSpace γ} {_ :
(hg.mono' hf.absolutelyContinuous).comp_measurable hf.measurable
#align measure_theory.ae_strongly_measurable.comp_quasi_measure_preserving MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
+theorem comp_measurePreserving {γ : Type _} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
+ {f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
+ (hf : MeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ :=
+ hg.comp_quasiMeasurePreserving hf.quasiMeasurePreserving
+
theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t := by
refine' ⟨range (hf.mk f), hf.stronglyMeasurable_mk.isSeparable_range, _⟩
This PR is the result of running
find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;
which firstly replaces .
focusing dots with ·
and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.
@@ -865,8 +865,7 @@ theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g
· rcases hx with ⟨y, rfl⟩
simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using
hf.tendsto_approx y
- ·
- simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using
+ · simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using
hg'.tendsto_approx x
#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend
@@ -1836,8 +1836,7 @@ theorem _root_.StronglyMeasurable.apply_continuousLinearMap {_m : MeasurableSpac
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
(ContinuousLinearMap.apply 𝕜 E v).continuous.comp_aestronglyMeasurable hφ
-#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map
-MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
+#align measure_theory.ae_strongly_measurable.apply_continuous_linear_map MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
theorem _root_.ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
{g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
@@ -1896,8 +1895,7 @@ protected noncomputable def mk (f : α → β) (hf : AEFinStronglyMeasurable f
theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
FinStronglyMeasurable (hf.mk f) μ :=
hf.choose_spec.1
-#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk
-MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
+#align measure_theory.ae_fin_strongly_measurable.fin_strongly_measurable_mk MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
@@ -967,10 +967,10 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
(hm : m ≤ m0) (hf : StronglyMeasurable[m] f) (μ : Measure α) [SigmaFinite (μ.trim hm)] :
∃ s : ℕ → Set α,
(∀ n, MeasurableSet[m] (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ n) ∧
- (⋃ i, s i) = Set.univ := by
+ ⋃ i, s i = Set.univ := by
let sigma_finite_sets := spanningSets (μ.trim hm)
let norm_sets := fun n : ℕ => { x | ‖f x‖ ≤ n }
- have norm_sets_spanning : (⋃ n, norm_sets n) = Set.univ := by
+ have norm_sets_spanning : ⋃ n, norm_sets n = Set.univ := by
ext1 x
simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
exact ⟨⌈‖f x‖⌉₊, Nat.le_ceil ‖f x‖⟩
@@ -985,7 +985,7 @@ theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0
refine' ⟨sets, fun n => ⟨h_meas n, h_finite n, _⟩, _⟩
· exact fun x hx => hx.2
· have :
- (⋃ i, sigma_finite_sets i ∩ norm_sets i) = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i := by
+ ⋃ i, sigma_finite_sets i ∩ norm_sets i = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i := by
refine' Set.iUnion_inter_of_monotone (monotone_spanningSets (μ.trim hm)) fun i j hij x => _
simp only [Set.mem_setOf_eq]
refine' fun hif => hif.trans _
@@ -812,7 +812,7 @@ theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableS
[TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
(h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
StronglyMeasurable f :=
- stronglyMeasurable_of_stronglyMeasurable_union_cover s (sᶜ) hs hs.compl (union_compl_self s).ge h₁
+ stronglyMeasurable_of_stronglyMeasurable_union_cover s sᶜ hs hs.compl (union_compl_self s).ge h₁
h₂
#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl
@@ -1062,7 +1062,7 @@ theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
· rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,
Set.compl_inter_self t, Set.empty_union]
exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n)
- · rw [← Set.union_iUnion (tᶜ) T]
+ · rw [← Set.union_iUnion tᶜ T]
exact Set.compl_union_self _
#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
@@ -1535,7 +1535,7 @@ theorem _root_.aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : M
refine' ⟨indicator s (h.mk f), h.stronglyMeasurable_mk.indicator hs, _⟩
have A : s.indicator f =ᵐ[μ.restrict s] s.indicator (h.mk f) :=
(indicator_ae_eq_restrict hs).trans (h.ae_eq_mk.trans <| (indicator_ae_eq_restrict hs).symm)
- have B : s.indicator f =ᵐ[μ.restrict (sᶜ)] s.indicator (h.mk f) :=
+ have B : s.indicator f =ᵐ[μ.restrict sᶜ] s.indicator (h.mk f) :=
(indicator_ae_eq_restrict_compl hs).trans (indicator_ae_eq_restrict_compl hs).symm
exact ae_of_ae_restrict_of_ae_restrict_compl _ A B
#align ae_strongly_measurable_indicator_iff aestronglyMeasurable_indicator_iff
@@ -1975,7 +1975,7 @@ end Order
variable [Zero β] [T2Space β]
theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
- ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict (tᶜ)] 0 ∧ SigmaFinite (μ.restrict t) := by
+ ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict tᶜ] 0 ∧ SigmaFinite (μ.restrict t) := by
rcases hf with ⟨g, hg, hfg⟩
obtain ⟨t, ht, hgt_zero, htμ⟩ := hg.exists_set_sigmaFinite
refine' ⟨t, ht, _, htμ⟩
@@ -1995,7 +1995,7 @@ protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
#align measure_theory.ae_fin_strongly_measurable.measurable_set MeasureTheory.AEFinStronglyMeasurable.measurableSet
theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
- f =ᵐ[μ.restrict (hf.sigmaFiniteSetᶜ)] 0 :=
+ f =ᵐ[μ.restrict hf.sigmaFiniteSetᶜ] 0 :=
hf.exists_set_sigmaFinite.choose_spec.2.1
#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_compl MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
measurability?
tactic (#5427)
This PR adds aesop tags to a few lemmas pertaining to strong measurability, allowing to prove e.g. StronglyMeasurable Real.log
using the measurability
tactic.
It also implements measurability?
via aesop?
.
Co-authored-by: Frédéric Dupuis <31101893+dupuisf@users.noreply.github.com>
@@ -128,7 +128,7 @@ open MeasureTheory
/-! ## Strongly measurable functions -/
-
+@[aesop 30% apply (rule_sets [Measurable])]
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {_ : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
AEStronglyMeasurable f μ :=
@@ -346,6 +346,7 @@ theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
/-- If the measure is sigma-finite, all strongly measurable functions are
`FinStronglyMeasurable`. -/
+@[aesop 5% apply (rule_sets [Measurable])]
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
(hf : StronglyMeasurable f) (μ : Measure α) [SigmaFinite μ] : FinStronglyMeasurable f μ :=
hf.finStronglyMeasurable_of_set_sigmaFinite MeasurableSet.univ (by simp)
@@ -353,6 +354,7 @@ protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : Me
#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable
/-- A strongly measurable function is measurable. -/
+@[aesop 5% apply (rule_sets [Measurable])]
protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
[MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f :=
measurable_of_tendsto_metrizable (fun n => (hf.approx n).measurable)
@@ -360,6 +362,7 @@ protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [Pse
#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable
/-- A strongly measurable function is almost everywhere measurable. -/
+@[aesop 5% apply (rule_sets [Measurable])]
protected theorem aemeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
[PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
(hf : StronglyMeasurable f) : AEMeasurable f μ :=
@@ -419,42 +422,42 @@ section Arithmetic
variable {mα : MeasurableSpace α} [TopologicalSpace β]
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f * g) :=
⟨fun n => hf.approx n * hg.approx n, fun x => (hf.tendsto_approx x).mul (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.mul MeasureTheory.StronglyMeasurable.mul
#align measure_theory.strongly_measurable.add MeasureTheory.StronglyMeasurable.add
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
StronglyMeasurable fun x => f x * c :=
hf.mul stronglyMeasurable_const
#align measure_theory.strongly_measurable.mul_const MeasureTheory.StronglyMeasurable.mul_const
#align measure_theory.strongly_measurable.add_const MeasureTheory.StronglyMeasurable.add_const
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
StronglyMeasurable fun x => c * f x :=
stronglyMeasurable_const.mul hf
#align measure_theory.strongly_measurable.const_mul MeasureTheory.StronglyMeasurable.const_mul
#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem inv [Group β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
StronglyMeasurable f⁻¹ :=
⟨fun n => (hf.approx n)⁻¹, fun x => (hf.tendsto_approx x).inv⟩
#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv
#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f / g) :=
⟨fun n => hf.approx n / hg.approx n, fun x => (hf.tendsto_approx x).div' (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.div MeasureTheory.StronglyMeasurable.div
#align measure_theory.strongly_measurable.sub MeasureTheory.StronglyMeasurable.sub
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => f x • g x :=
@@ -462,17 +465,19 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
#align measure_theory.strongly_measurable.smul MeasureTheory.StronglyMeasurable.smul
#align measure_theory.strongly_measurable.vadd MeasureTheory.StronglyMeasurable.vadd
+@[measurability]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable (c • f) :=
⟨fun n => c • hf.approx n, fun x => (hf.tendsto_approx x).const_smul c⟩
#align measure_theory.strongly_measurable.const_smul MeasureTheory.StronglyMeasurable.const_smul
+@[measurability]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
(c : 𝕜) : StronglyMeasurable fun x => c • f x :=
hf.const_smul c
#align measure_theory.strongly_measurable.const_smul' MeasureTheory.StronglyMeasurable.const_smul'
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
(hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c :=
continuous_smul.comp_stronglyMeasurable (hf.prod_mk stronglyMeasurable_const)
@@ -516,12 +521,14 @@ open Filter
open Filter
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) :=
⟨fun n => hf.approx n ⊔ hg.approx n, fun x =>
(hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩
#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
(hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) :=
⟨fun n => hf.approx n ⊓ hg.approx n, fun x =>
@@ -539,7 +546,7 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod := by
induction' l with f l ihl; · exact stronglyMeasurable_one
@@ -549,7 +556,7 @@ theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'
#align list.strongly_measurable_sum' List.stronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod (l : List (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) :
StronglyMeasurable fun x => (l.map fun f : α → M => f x).prod := by
@@ -563,7 +570,7 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α}
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod := by
rcases l with ⟨l⟩
@@ -571,7 +578,7 @@ theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'
#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, StronglyMeasurable f) :
StronglyMeasurable fun x => (s.map fun f : α → M => f x).prod := by
@@ -579,14 +586,14 @@ theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prod
#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i in s, f i) :=
Finset.prod_induction _ _ (fun _a _b ha hb => ha.mul hb) (@stronglyMeasurable_one α M _ _ _) hf
#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'
#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i in s, f i a := by
simpa only [← Finset.prod_apply] using s.stronglyMeasurable_prod' hf
@@ -620,6 +627,7 @@ section SecondCountableStronglyMeasurable
variable {mα : MeasurableSpace α} [MeasurableSpace β]
/-- In a space with second countable topology, measurable implies strongly measurable. -/
+@[aesop 90% apply (rule_sets [Measurable])]
theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
[SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
StronglyMeasurable f := by
@@ -638,6 +646,7 @@ theorem _root_.stronglyMeasurable_iff_measurable [TopologicalSpace β] [Metrizab
⟨fun h => h.measurable, fun h => Measurable.stronglyMeasurable h⟩
#align strongly_measurable_iff_measurable stronglyMeasurable_iff_measurable
+@[measurability]
theorem _root_.stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
[OpensMeasurableSpace α] [SecondCountableTopology α] : StronglyMeasurable (id : α → α) :=
measurable_id.stronglyMeasurable
@@ -807,38 +816,45 @@ theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableS
h₂
#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl
+@[measurability]
protected theorem indicator {_ : MeasurableSpace α} [TopologicalSpace β] [Zero β]
(hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
StronglyMeasurable (s.indicator f) :=
hf.piecewise hs stronglyMeasurable_const
#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator
+@[aesop safe 20 apply (rule_sets [Measurable])]
protected theorem dist {_ : MeasurableSpace α} {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
StronglyMeasurable fun x => dist (f x) (g x) :=
continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)
#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist
+@[measurability]
protected theorem norm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ :=
continuous_norm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm
+@[measurability]
protected theorem nnnorm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ :=
continuous_nnnorm.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm
+@[measurability]
protected theorem ennnorm {_ : MeasurableSpace α} {β : Type _} [SeminormedAddCommGroup β]
{f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
(ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).measurable
#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm
+@[measurability]
protected theorem real_toNNReal {_ : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => (f x).toNNReal :=
continuous_real_toNNReal.comp_stronglyMeasurable hf
#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal
+@[measurability]
theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
{mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
(hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
@@ -1020,6 +1036,8 @@ protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop
end sequence
+/-- A finitely strongly measurable function is strongly measurable. -/
+@[aesop 5% apply (rule_sets [Measurable])]
protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
(hf : FinStronglyMeasurable f μ) : StronglyMeasurable f :=
⟨hf.approx, hf.tendsto_approx⟩
@@ -1058,6 +1076,7 @@ section Arithmetic
variable [TopologicalSpace β]
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ := by
refine'
@@ -1067,6 +1086,7 @@ protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMe
exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)
#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ :=
⟨fun n => hf.approx n + hg.approx n, fun n =>
@@ -1076,6 +1096,7 @@ protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasura
fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add
+@[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
FinStronglyMeasurable (-f) μ := by
refine' ⟨fun n => -hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).neg⟩
@@ -1084,6 +1105,7 @@ protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMe
exact hf.fin_support_approx n
#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg
+@[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ :=
⟨fun n => hf.approx n - hg.approx n, fun n =>
@@ -1093,6 +1115,7 @@ protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurab
fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)⟩
#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub
+@[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
FinStronglyMeasurable (c • f) μ := by
@@ -1107,6 +1130,7 @@ section Order
variable [TopologicalSpace β] [Zero β]
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ := by
refine'
@@ -1116,6 +1140,7 @@ protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMe
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩
#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
(hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ := by
refine'
@@ -1146,13 +1171,13 @@ theorem aefinStronglyMeasurable_zero {α β} {_ : MeasurableSpace α} (μ : Meas
/-! ## Almost everywhere strongly measurable functions -/
-
+@[measurability]
theorem aestronglyMeasurable_const {α β} {_ : MeasurableSpace α} {μ : Measure α}
[TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun _ : α => b) μ :=
stronglyMeasurable_const.aestronglyMeasurable
#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_const
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem aestronglyMeasurable_one {α β} {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
[One β] : AEStronglyMeasurable (1 : α → β) μ :=
stronglyMeasurable_one.aestronglyMeasurable
@@ -1179,6 +1204,7 @@ theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace
exact ⟨fun _ => f default, stronglyMeasurable_const, rfl⟩
#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measure
+@[measurability]
theorem SimpleFunc.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
(f : α →ₛ β) : AEStronglyMeasurable f μ :=
f.stronglyMeasurable.aestronglyMeasurable
@@ -1210,6 +1236,7 @@ theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
hf.choose_spec.2
#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk
+@[aesop 5% apply (rule_sets [Measurable])]
protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEStronglyMeasurable f μ) :
AEMeasurable f μ :=
@@ -1283,7 +1310,7 @@ theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Me
section Arithmetic
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.stronglyMeasurable_mk.mul hg.stronglyMeasurable_mk,
@@ -1291,28 +1318,28 @@ protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f
#align measure_theory.ae_strongly_measurable.mul MeasureTheory.AEStronglyMeasurable.mul
#align measure_theory.ae_strongly_measurable.add MeasureTheory.AEStronglyMeasurable.add
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
AEStronglyMeasurable (fun x => f x * c) μ :=
hf.mul aestronglyMeasurable_const
#align measure_theory.ae_strongly_measurable.mul_const MeasureTheory.AEStronglyMeasurable.mul_const
#align measure_theory.ae_strongly_measurable.add_const MeasureTheory.AEStronglyMeasurable.add_const
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
AEStronglyMeasurable (fun x => c * f x) μ :=
aestronglyMeasurable_const.mul hf
#align measure_theory.ae_strongly_measurable.const_mul MeasureTheory.AEStronglyMeasurable.const_mul
#align measure_theory.ae_strongly_measurable.const_add MeasureTheory.AEStronglyMeasurable.const_add
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem inv [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable f⁻¹ μ :=
⟨(hf.mk f)⁻¹, hf.stronglyMeasurable_mk.inv, hf.ae_eq_mk.inv⟩
#align measure_theory.ae_strongly_measurable.inv MeasureTheory.AEStronglyMeasurable.inv
#align measure_theory.ae_strongly_measurable.neg MeasureTheory.AEStronglyMeasurable.neg
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ :=
⟨hf.mk f / hg.mk g, hf.stronglyMeasurable_mk.div hg.stronglyMeasurable_mk,
@@ -1320,7 +1347,7 @@ protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurabl
#align measure_theory.ae_strongly_measurable.div MeasureTheory.AEStronglyMeasurable.div
#align measure_theory.ae_strongly_measurable.sub MeasureTheory.AEStronglyMeasurable.sub
-@[to_additive]
+@[to_additive (attr := aesop safe 20 apply (rule_sets [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
{g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => f x • g x) μ :=
@@ -1328,17 +1355,19 @@ protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [Continuous
#align measure_theory.ae_strongly_measurable.smul MeasureTheory.AEStronglyMeasurable.smul
#align measure_theory.ae_strongly_measurable.vadd MeasureTheory.AEStronglyMeasurable.vadd
+@[measurability]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
(hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ :=
⟨c • hf.mk f, hf.stronglyMeasurable_mk.const_smul c, hf.ae_eq_mk.const_smul c⟩
#align measure_theory.ae_strongly_measurable.const_smul MeasureTheory.AEStronglyMeasurable.const_smul
+@[measurability]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
(hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ :=
hf.const_smul c
#align measure_theory.ae_strongly_measurable.const_smul' MeasureTheory.AEStronglyMeasurable.const_smul'
-@[to_additive]
+@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
(hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ :=
continuous_smul.comp_aestronglyMeasurable (hf.prod_mk aestronglyMeasurable_const)
@@ -1349,12 +1378,14 @@ end Arithmetic
section Order
+@[aesop safe 20 apply (rule_sets [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.stronglyMeasurable_mk.sup hg.stronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_strongly_measurable.sup MeasureTheory.AEStronglyMeasurable.sup
+@[aesop safe 20 apply (rule_sets [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.stronglyMeasurable_mk.inf hg.stronglyMeasurable_mk,
@@ -1372,7 +1403,7 @@ section Monoid
variable {M : Type _} [Monoid M] [TopologicalSpace M] [ContinuousMul M]
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ := by
induction' l with f l ihl; · exact aestronglyMeasurable_one
@@ -1382,7 +1413,7 @@ theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
#align list.ae_strongly_measurable_prod' List.aestronglyMeasurable_prod'
#align list.ae_strongly_measurable_sum' List.aestronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod
(l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ := by
@@ -1396,7 +1427,7 @@ section CommMonoid
variable {M : Type _} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ := by
rcases l with ⟨l⟩
@@ -1404,7 +1435,7 @@ theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
#align multiset.ae_strongly_measurable_prod' Multiset.aestronglyMeasurable_prod'
#align multiset.ae_strongly_measurable_sum' Multiset.aestronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
(hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ := by
@@ -1412,7 +1443,7 @@ theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
#align multiset.ae_strongly_measurable_prod Multiset.aestronglyMeasurable_prod
#align multiset.ae_strongly_measurable_sum Multiset.aestronglyMeasurable_sum
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i in s, f i) μ :=
Multiset.aestronglyMeasurable_prod' _ fun _g hg =>
@@ -1421,7 +1452,7 @@ theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type _} {f : ι → α
#align finset.ae_strongly_measurable_prod' Finset.aestronglyMeasurable_prod'
#align finset.ae_strongly_measurable_sum' Finset.aestronglyMeasurable_sum'
-@[to_additive]
+@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type _} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
AEStronglyMeasurable (fun a => ∏ i in s, f i a) μ := by
@@ -1436,11 +1467,13 @@ section SecondCountableAEStronglyMeasurable
variable [MeasurableSpace β]
/-- In a space with second countable topology, measurable implies strongly measurable. -/
+@[aesop 90% apply (rule_sets [Measurable])]
theorem _root_.AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
[SecondCountableTopology β] (hf : AEMeasurable f μ) : AEStronglyMeasurable f μ :=
⟨hf.mk f, hf.measurable_mk.stronglyMeasurable, hf.ae_eq_mk⟩
#align ae_measurable.ae_strongly_measurable AEMeasurable.aestronglyMeasurable
+@[measurability]
theorem _root_.aestronglyMeasurable_id {α : Type _} [TopologicalSpace α] [PseudoMetrizableSpace α]
{_ : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
AEStronglyMeasurable (id : α → α) μ :=
@@ -1455,33 +1488,39 @@ theorem _root_.aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β]
end SecondCountableAEStronglyMeasurable
+@[aesop safe 20 apply (rule_sets [Measurable])]
protected theorem dist {β : Type _} [PseudoMetricSpace β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun x => dist (f x) (g x)) μ :=
continuous_dist.comp_aestronglyMeasurable (hf.prod_mk hg)
#align measure_theory.ae_strongly_measurable.dist MeasureTheory.AEStronglyMeasurable.dist
+@[measurability]
protected theorem norm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖) μ :=
continuous_norm.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.norm MeasureTheory.AEStronglyMeasurable.norm
+@[measurability]
protected theorem nnnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ :=
continuous_nnnorm.comp_aestronglyMeasurable hf
#align measure_theory.ae_strongly_measurable.nnnorm MeasureTheory.AEStronglyMeasurable.nnnorm
+@[measurability]
protected theorem ennnorm {β : Type _} [SeminormedAddCommGroup β] {f : α → β}
(hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
(ENNReal.continuous_coe.comp_aestronglyMeasurable hf.nnnorm).aemeasurable
#align measure_theory.ae_strongly_measurable.ennnorm MeasureTheory.AEStronglyMeasurable.ennnorm
+@[aesop safe 20 apply (rule_sets [Measurable])]
protected theorem edist {β : Type _} [SeminormedAddCommGroup β] {f g : α → β}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
AEMeasurable (fun a => edist (f a) (g a)) μ :=
(continuous_edist.comp_aestronglyMeasurable (hf.prod_mk hg)).aemeasurable
#align measure_theory.ae_strongly_measurable.edist MeasureTheory.AEStronglyMeasurable.edist
+@[measurability]
protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (f x).toNNReal) μ :=
continuous_real_toNNReal.comp_aestronglyMeasurable hf
@@ -1501,6 +1540,7 @@ theorem _root_.aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : M
exact ae_of_ae_restrict_of_ae_restrict_compl _ A B
#align ae_strongly_measurable_indicator_iff aestronglyMeasurable_indicator_iff
+@[measurability]
protected theorem indicator [Zero β] (hfm : AEStronglyMeasurable f μ) {s : Set α}
(hs : MeasurableSet s) : AEStronglyMeasurable (s.indicator f) μ :=
(aestronglyMeasurable_indicator_iff hs).mpr hfm.restrict
@@ -1694,12 +1734,14 @@ theorem _root_.aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {
rfl
#align ae_strongly_measurable_add_measure_iff aestronglyMeasurable_add_measure_iff
+@[measurability]
theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
(hμ : AEStronglyMeasurable f μ) (hν : AEStronglyMeasurable f ν) :
AEStronglyMeasurable f (μ + ν) :=
aestronglyMeasurable_add_measure_iff.2 ⟨hμ, hν⟩
#align measure_theory.ae_strongly_measurable.add_measure MeasureTheory.AEStronglyMeasurable.add_measure
+@[measurability]
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
(h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) :=
@@ -1730,6 +1772,7 @@ theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β
#align measure_theory.ae_strongly_measurable.ae_strongly_measurable_uIoc_iff
MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
+@[measurability]
theorem smul_measure {R : Type _} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) :=
⟨h.mk f, h.stronglyMeasurable_mk, ae_smul_measure h.ae_eq_mk c⟩
@@ -1789,6 +1832,7 @@ theorem _root_.StronglyMeasurable.apply_continuousLinearMap {_m : MeasurableSpac
(ContinuousLinearMap.apply 𝕜 E v).continuous.comp_stronglyMeasurable hφ
#align strongly_measurable.apply_continuous_linear_map StronglyMeasurable.apply_continuousLinearMap
+@[measurability]
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
AEStronglyMeasurable (fun a => φ a v) μ :=
(ContinuousLinearMap.apply 𝕜 E v).continuous.comp_aestronglyMeasurable hφ
@@ -1860,6 +1904,7 @@ theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f :=
#align measure_theory.ae_fin_strongly_measurable.ae_eq_mk
MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
+@[aesop 10% apply (rule_sets [Measurable])]
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
[PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
AEMeasurable f μ :=
@@ -1871,29 +1916,34 @@ end Mk
section Arithmetic
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ :=
⟨hf.mk f * hg.mk g, hf.finStronglyMeasurable_mk.mul hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.mul hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.mul MeasureTheory.AEFinStronglyMeasurable.mul
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ :=
⟨hf.mk f + hg.mk g, hf.finStronglyMeasurable_mk.add hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.add hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.add MeasureTheory.AEFinStronglyMeasurable.add
+@[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
AEFinStronglyMeasurable (-f) μ :=
⟨-hf.mk f, hf.finStronglyMeasurable_mk.neg, hf.ae_eq_mk.neg⟩
#align measure_theory.ae_fin_strongly_measurable.neg MeasureTheory.AEFinStronglyMeasurable.neg
+@[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ :=
⟨hf.mk f - hg.mk g, hf.finStronglyMeasurable_mk.sub hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sub hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sub MeasureTheory.AEFinStronglyMeasurable.sub
+@[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
[DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
AEFinStronglyMeasurable (c • f) μ :=
@@ -1906,12 +1956,14 @@ section Order
variable [Zero β]
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ :=
⟨hf.mk f ⊔ hg.mk g, hf.finStronglyMeasurable_mk.sup hg.finStronglyMeasurable_mk,
hf.ae_eq_mk.sup hg.ae_eq_mk⟩
#align measure_theory.ae_fin_strongly_measurable.sup MeasureTheory.AEFinStronglyMeasurable.sup
+@[aesop safe 20 (rule_sets [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
(hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ :=
⟨hf.mk f ⊓ hg.mk g, hf.finStronglyMeasurable_mk.inf hg.finStronglyMeasurable_mk,
@@ -1967,6 +2019,13 @@ theorem finStronglyMeasurable_iff_measurable {_m0 : MeasurableSpace α} (μ : Me
⟨fun h => h.measurable, fun h => (Measurable.stronglyMeasurable h).finStronglyMeasurable μ⟩
#align measure_theory.fin_strongly_measurable_iff_measurable MeasureTheory.finStronglyMeasurable_iff_measurable
+/-- In a space with second countable topology and a sigma-finite measure, a measurable function
+is `FinStronglyMeasurable`. -/
+@[aesop 90% apply (rule_sets [Measurable])]
+theorem finStronglyMeasurable_of_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
+ [SigmaFinite μ] (hf : Measurable f) : FinStronglyMeasurable f μ :=
+ (finStronglyMeasurable_iff_measurable μ).mpr hf
+
/-- In a space with second countable topology and a sigma-finite measure,
`AEFinStronglyMeasurable` and `AEMeasurable` are equivalent. -/
theorem aefinStronglyMeasurable_iff_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
@@ -1974,6 +2033,13 @@ theorem aefinStronglyMeasurable_iff_aemeasurable {_m0 : MeasurableSpace α} (μ
simp_rw [AEFinStronglyMeasurable, AEMeasurable, finStronglyMeasurable_iff_measurable]
#align measure_theory.ae_fin_strongly_measurable_iff_ae_measurable MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable
+/-- In a space with second countable topology and a sigma-finite measure,
+ an `AEMeasurable` function is `AEFinStronglyMeasurable`. -/
+@[aesop 90% apply (rule_sets [Measurable])]
+theorem aefinStronglyMeasurable_of_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
+ [SigmaFinite μ] (hf : AEMeasurable f μ) : AEFinStronglyMeasurable f μ :=
+ (aefinStronglyMeasurable_iff_aemeasurable μ).mpr hf
+
end SecondCountableTopology
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type _} [TopologicalSpace ι]
@@ -1003,7 +1003,7 @@ section sequence
variable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ)
-/-- A sequence of simple functions such that `∀ x, Tendsto (λ n, hf.approx n x) atTop (𝓝 (f x))`
+/-- A sequence of simple functions such that `∀ x, Tendsto (fun n ↦ hf.approx n x) atTop (𝓝 (f x))`
and `∀ n, μ (support (hf.approx n)) < ∞`. These properties are given by
`FinStronglyMeasurable.tendsto_approx` and `FinStronglyMeasurable.fin_support_approx`. -/
protected noncomputable def approx : ℕ → α →ₛ β :=
@@ -1652,7 +1652,7 @@ one can select a strongly measurable function as the almost everywhere limit. -/
theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
{f : ℕ → α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
- ∃ (f_lim : α → β)(hf_lim_meas : StronglyMeasurable f_lim),
+ ∃ (f_lim : α → β) (hf_lim_meas : StronglyMeasurable f_lim),
∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) := by
borelize β
obtain ⟨g, _, hg⟩ :
This PR also uncomment assert_not_exists
which was forgotten to be uncommented when assert_not_exists
was ported.
Co-authored-by: Komyyy <pol_tta@outlook.jp>
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
! This file was ported from Lean 3 source module measure_theory.function.strongly_measurable.basic
-! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
+! leanprover-community/mathlib commit ef95945cd48c932c9e034872bd25c3c220d9c946
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -483,20 +483,27 @@ end Arithmetic
section MulAction
-variable [TopologicalSpace β] {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable {M G G₀ : Type _}
+variable [TopologicalSpace β]
+variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
+variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
theorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iff
-variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+nonrec theorem _root_.IsUnit.stronglyMeasurable_const_smul_iff {_ : MeasurableSpace α} {c : M}
+ (hc : IsUnit c) :
+ (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
+ let ⟨u, hu⟩ := hc
+ hu ▸ stronglyMeasurable_const_smul_iff u
+#align is_unit.strongly_measurable_const_smul_iff IsUnit.stronglyMeasurable_const_smul_iff
-theorem _root_.stronglyMeasurable_const_smul_iff₀ {m : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
- (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f := by
- refine' ⟨fun h => _, fun h => h.const_smul c⟩
- convert h.const_smul' c⁻¹
- simp [smul_smul, inv_mul_cancel hc]
+theorem _root_.stronglyMeasurable_const_smul_iff₀ {_ : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
+ (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
+ (IsUnit.mk0 _ hc).stronglyMeasurable_const_smul_iff
#align strongly_measurable_const_smul_iff₀ stronglyMeasurable_const_smul_iff₀
end MulAction
@@ -1743,20 +1750,25 @@ end NormedSpace
section MulAction
-variable {G : Type _} [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable {M G G₀ : Type _}
+variable [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
+variable [Group G] [MulAction G β] [ContinuousConstSMul G β]
+variable [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
theorem _root_.aestronglyMeasurable_const_smul_iff (c : G) :
AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
#align ae_strongly_measurable_const_smul_iff aestronglyMeasurable_const_smul_iff
-variable {G₀ : Type _} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
+nonrec theorem _root_.IsUnit.aestronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
+ AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
+ let ⟨u, hu⟩ := hc
+ hu ▸ aestronglyMeasurable_const_smul_iff u
+#align is_unit.ae_strongly_measurable_const_smul_iff IsUnit.aestronglyMeasurable_const_smul_iff
theorem _root_.aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
- AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ := by
- refine' ⟨fun h => _, fun h => h.const_smul c⟩
- convert h.const_smul' c⁻¹
- simp [smul_smul, inv_mul_cancel hc]
+ AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ :=
+ (IsUnit.mk0 _ hc).aestronglyMeasurable_const_smul_iff
#align ae_strongly_measurable_const_smul_iff₀ aestronglyMeasurable_const_smul_iff₀
end MulAction
@@ -2039,4 +2051,4 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
end MeasureTheory
-- Guard against import creep
--- assert_not_exists inner_product_space
+assert_not_exists InnerProductSpace
The unported dependencies are
algebra.order.module
init.core
linear_algebra.free_module.finite.rank
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
linear_algebra.free_module.rank
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file