Set integral #
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In this file we prove some properties of ∫ x in s, f x ∂μ. Recall that this notation
is defined as ∫ x, f x ∂(μ.restrict s). In integral_indicator we prove that for a measurable
function f and a measurable set s this definition coincides with another natural definition:
∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ, where indicator s f x is equal to f x for x ∈ s
and is zero otherwise.
Since ∫ x in s, f x ∂μ is a notation, one can rewrite or apply any theorem about ∫ x, f x ∂μ
directly. In this file we prove some theorems about dependence of ∫ x in s, f x ∂μ on s, e.g.
integral_union, integral_empty, integral_univ.
We use the property integrable_on f s μ := integrable f (μ.restrict s), defined in
measure_theory.integrable_on. We also defined in that same file a predicate
integrable_at_filter (f : α → E) (l : filter α) (μ : measure α) saying that f is integrable at
some set s ∈ l.
Finally, we prove a version of the
Fundamental theorem of calculus
for set integral, see filter.tendsto.integral_sub_linear_is_o_ae and its corollaries.
Namely, consider a measurably generated filter l, a measure μ finite at this filter, and
a function f that has a finite limit c at l ⊓ μ.ae. Then ∫ x in s, f x ∂μ = μ s • c + o(μ s)
as s tends to l.small_sets, i.e. for any ε>0 there exists t ∈ l such that
‖∫ x in s, f x ∂μ - μ s • c‖ ≤ ε * μ s whenever s ⊆ t. We also formulate a version of this
theorem for a locally finite measure μ and a function f continuous at a point a.
Notation #
We provide the following notations for expressing the integral of a function on a set :
∫ a in s, f a ∂μismeasure_theory.integral (μ.restrict s) f∫ a in s, f ais∫ a in s, f a ∂volume
Note that the set notations are defined in the file measure_theory/integral/bochner,
but we reference them here because all theorems about set integrals are in this file.
theorem
measure_theory.set_integral_congr_ae₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f g : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measure_theory.null_measurable_set s μ)
(h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x) :
theorem
measure_theory.set_integral_congr_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f g : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measurable_set s)
(h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x) :
theorem
measure_theory.set_integral_congr₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f g : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measure_theory.null_measurable_set s μ)
(h : set.eq_on f g s) :
theorem
measure_theory.set_integral_congr
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f g : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measurable_set s)
(h : set.eq_on f g s) :
theorem
measure_theory.set_integral_congr_set_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hst : s =ᵐ[μ] t) :
theorem
measure_theory.integral_union_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hst : measure_theory.ae_disjoint μ s t)
(ht : measure_theory.null_measurable_set t μ)
(hfs : measure_theory.integrable_on f s μ)
(hft : measure_theory.integrable_on f t μ) :
theorem
measure_theory.integral_union
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hst : disjoint s t)
(ht : measurable_set t)
(hfs : measure_theory.integrable_on f s μ)
(hft : measure_theory.integrable_on f t μ) :
theorem
measure_theory.integral_diff
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measurable_set t)
(hfs : measure_theory.integrable_on f s μ)
(hts : t ⊆ s) :
theorem
measure_theory.integral_inter_add_diff₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measure_theory.null_measurable_set t μ)
(hfs : measure_theory.integrable_on f s μ) :
theorem
measure_theory.integral_inter_add_diff
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measurable_set t)
(hfs : measure_theory.integrable_on f s μ) :
theorem
measure_theory.integral_finset_bUnion
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
(t : finset ι)
{s : ι → set α}
(hs : ∀ (i : ι), i ∈ t → measurable_set (s i))
(h's : ↑t.pairwise (disjoint on s))
(hf : ∀ (i : ι), i ∈ t → measure_theory.integrable_on f (s i) μ) :
theorem
measure_theory.integral_fintype_Union
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[fintype ι]
{s : ι → set α}
(hs : ∀ (i : ι), measurable_set (s i))
(h's : pairwise (disjoint on s))
(hf : ∀ (i : ι), measure_theory.integrable_on f (s i) μ) :
theorem
measure_theory.integral_empty
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E] :
theorem
measure_theory.integral_univ
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E] :
theorem
measure_theory.integral_add_compl₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measure_theory.null_measurable_set s μ)
(hfi : measure_theory.integrable f μ) :
theorem
measure_theory.integral_add_compl
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measurable_set s)
(hfi : measure_theory.integrable f μ) :
theorem
measure_theory.integral_indicator
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hs : measurable_set s) :
For a function f and a measurable set s, the integral of indicator s f
over the whole space is equal to ∫ x in s, f x ∂μ defined as ∫ x, f x ∂(μ.restrict s).
theorem
measure_theory.set_integral_indicator
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measurable_set t) :
theorem
measure_theory.of_real_set_integral_one_of_measure_ne_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : ⇑μ s ≠ ⊤) :
theorem
measure_theory.of_real_set_integral_one
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(s : set α) :
theorem
measure_theory.integral_piecewise
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f g : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[decidable_pred (λ (_x : α), _x ∈ s)]
(hs : measurable_set s)
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g sᶜ μ) :
theorem
measure_theory.tendsto_set_integral_of_monotone
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[countable ι]
[semilattice_sup ι]
{s : ι → set α}
(hsm : ∀ (i : ι), measurable_set (s i))
(h_mono : monotone s)
(hfi : measure_theory.integrable_on f (⋃ (n : ι), s n) μ) :
theorem
measure_theory.has_sum_integral_Union_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[countable ι]
{s : ι → set α}
(hm : ∀ (i : ι), measure_theory.null_measurable_set (s i) μ)
(hd : pairwise (measure_theory.ae_disjoint μ on s))
(hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
theorem
measure_theory.has_sum_integral_Union
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[countable ι]
{s : ι → set α}
(hm : ∀ (i : ι), measurable_set (s i))
(hd : pairwise (disjoint on s))
(hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
theorem
measure_theory.integral_Union
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[countable ι]
{s : ι → set α}
(hm : ∀ (i : ι), measurable_set (s i))
(hd : pairwise (disjoint on s))
(hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
theorem
measure_theory.integral_Union_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{ι : Type u_2}
[countable ι]
{s : ι → set α}
(hm : ∀ (i : ι), measure_theory.null_measurable_set (s i) μ)
(hd : pairwise (measure_theory.ae_disjoint μ on s))
(hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
theorem
measure_theory.set_integral_eq_zero_of_ae_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht_eq : ∀ᵐ (x : α) ∂μ, x ∈ t → f x = 0) :
theorem
measure_theory.set_integral_eq_zero_of_forall_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht_eq : ∀ (x : α), x ∈ t → f x = 0) :
theorem
measure_theory.integral_union_eq_left_of_ae_aux
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht_eq : ∀ᵐ (x : α) ∂μ.restrict t, f x = 0)
(haux : measure_theory.strongly_measurable f)
(H : measure_theory.integrable_on f (s ∪ t) μ) :
theorem
measure_theory.integral_union_eq_left_of_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht_eq : ∀ᵐ (x : α) ∂μ.restrict t, f x = 0) :
theorem
measure_theory.integral_union_eq_left_of_forall₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → E}
(ht : measure_theory.null_measurable_set t μ)
(ht_eq : ∀ (x : α), x ∈ t → f x = 0) :
theorem
measure_theory.integral_union_eq_left_of_forall
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → E}
(ht : measurable_set t)
(ht_eq : ∀ (x : α), x ∈ t → f x = 0) :
theorem
measure_theory.set_integral_eq_of_subset_of_ae_diff_eq_zero_aux
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(hts : s ⊆ t)
(h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0)
(haux : measure_theory.strongly_measurable f)
(h'aux : measure_theory.integrable_on f t μ) :
theorem
measure_theory.set_integral_eq_of_subset_of_ae_diff_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measure_theory.null_measurable_set t μ)
(hts : s ⊆ t)
(h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0) :
If a function vanishes almost everywhere on t \ s with s ⊆ t, then its integrals on s
and t coincide if t is null-measurable.
theorem
measure_theory.set_integral_eq_of_subset_of_forall_diff_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(ht : measurable_set t)
(hts : s ⊆ t)
(h't : ∀ (x : α), x ∈ t \ s → f x = 0) :
If a function vanishes on t \ s with s ⊆ t, then its integrals on s
and t coincide if t is measurable.
theorem
measure_theory.set_integral_eq_integral_of_ae_compl_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(h : ∀ᵐ (x : α) ∂μ, x ∉ s → f x = 0) :
If a function vanishes almost everywhere on sᶜ, then its integral on s
coincides with its integral on the whole space.
theorem
measure_theory.set_integral_eq_integral_of_forall_compl_eq_zero
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(h : ∀ (x : α), x ∉ s → f x = 0) :
If a function vanishes on sᶜ, then its integral on s coincides with its integral on the
whole space.
theorem
measure_theory.set_integral_neg_eq_set_integral_nonpos
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[linear_order E]
{f : α → E}
(hf : measure_theory.ae_strongly_measurable f μ) :
theorem
measure_theory.set_integral_const
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(c : E) :
@[simp]
theorem
measure_theory.integral_indicator_const
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
(e : E)
⦃s : set α⦄
(s_meas : measurable_set s) :
@[simp]
theorem
measure_theory.integral_indicator_one
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
⦃s : set α⦄
(hs : measurable_set s) :
theorem
measure_theory.set_integral_indicator_const_Lp
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{s t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{p : ennreal}
(hs : measurable_set s)
(ht : measurable_set t)
(hμt : ⇑μ t ≠ ⊤)
(x : E) :
theorem
measure_theory.integral_indicator_const_Lp
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{t : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{p : ennreal}
(ht : measurable_set t)
(hμt : ⇑μ t ≠ ⊤)
(x : E) :
theorem
measure_theory.set_integral_map
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{β : Type u_2}
[measurable_space β]
{g : α → β}
{f : β → E}
{s : set β}
(hs : measurable_set s)
(hf : measure_theory.ae_strongly_measurable f (measure_theory.measure.map g μ))
(hg : ae_measurable g μ) :
theorem
measurable_embedding.set_integral_map
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{β : Type u_2}
{_x : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(g : β → E)
(s : set β) :
theorem
closed_embedding.set_integral_map
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[topological_space α]
[borel_space α]
{β : Type u_2}
[measurable_space β]
[topological_space β]
[borel_space β]
{g : α → β}
{f : β → E}
(s : set β)
(hg : closed_embedding g) :
theorem
measure_theory.measure_preserving.set_integral_preimage_emb
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{β : Type u_2}
{_x : measurable_space β}
{f : α → β}
{ν : measure_theory.measure β . "volume_tac"}
(h₁ : measure_theory.measure_preserving f μ ν)
(h₂ : measurable_embedding f)
(g : β → E)
(s : set β) :
theorem
measure_theory.measure_preserving.set_integral_image_emb
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{β : Type u_2}
{_x : measurable_space β}
{f : α → β}
{ν : measure_theory.measure β . "volume_tac"}
(h₁ : measure_theory.measure_preserving f μ ν)
(h₂ : measurable_embedding f)
(g : β → E)
(s : set α) :
theorem
measure_theory.set_integral_map_equiv
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{β : Type u_2}
[measurable_space β]
(e : α ≃ᵐ β)
(f : β → E)
(s : set β) :
theorem
measure_theory.norm_set_integral_le_of_norm_le_const_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{C : ℝ}
(hs : ⇑μ s < ⊤)
(hC : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C) :
theorem
measure_theory.norm_set_integral_le_of_norm_le_const_ae'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{C : ℝ}
(hs : ⇑μ s < ⊤)
(hC : ∀ᵐ (x : α) ∂μ, x ∈ s → ‖f x‖ ≤ C)
(hfm : measure_theory.ae_strongly_measurable f (μ.restrict s)) :
theorem
measure_theory.norm_set_integral_le_of_norm_le_const_ae''
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{C : ℝ}
(hs : ⇑μ s < ⊤)
(hsm : measurable_set s)
(hC : ∀ᵐ (x : α) ∂μ, x ∈ s → ‖f x‖ ≤ C) :
theorem
measure_theory.norm_set_integral_le_of_norm_le_const
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{C : ℝ}
(hs : ⇑μ s < ⊤)
(hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C)
(hfm : measure_theory.ae_strongly_measurable f (μ.restrict s)) :
theorem
measure_theory.norm_set_integral_le_of_norm_le_const'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{s : set α}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{C : ℝ}
(hs : ⇑μ s < ⊤)
(hsm : measurable_set s)
(hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C) :
theorem
measure_theory.set_integral_pos_iff_support_of_nonneg_ae
{α : Type u_1}
[measurable_space α]
{s : set α}
{μ : measure_theory.measure α}
{f : α → ℝ}
(hf : 0 ≤ᵐ[μ.restrict s] f)
(hfi : measure_theory.integrable_on f s μ) :
theorem
measure_theory.set_integral_gt_gt
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{R : ℝ}
{f : α → ℝ}
(hR : 0 ≤ R)
(hfm : measurable f)
(hfint : measure_theory.integrable_on f {x : α | ↑R < f x} μ)
(hμ : ⇑μ {x : α | ↑R < f x} ≠ 0) :
theorem
measure_theory.set_integral_trim
{E : Type u_3}
[normed_add_comm_group E]
[complete_space E]
[normed_space ℝ E]
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
{f : α → E}
(hf_meas : measure_theory.strongly_measurable f)
{s : set α}
(hs : measurable_set s) :
Lemmas about adding and removing interval boundaries #
The primed lemmas take explicit arguments about the endpoint having zero measure, while the
unprimed ones use [has_no_atoms μ].
theorem
measure_theory.integral_Icc_eq_integral_Ioc'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.integral_Icc_eq_integral_Ico'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.integral_Ioc_eq_integral_Ioo'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.integral_Ico_eq_integral_Ioo'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.integral_Icc_eq_integral_Ioo'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0)
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.integral_Iic_eq_integral_Iio'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.integral_Ici_eq_integral_Ioi'
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.integral_Icc_eq_integral_Ioc
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Icc_eq_integral_Ico
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Ioc_eq_integral_Ioo
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Ico_eq_integral_Ioo
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Icc_eq_integral_Ioo
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a b : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Iic_eq_integral_Iio
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.integral_Ici_eq_integral_Ioi
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{f : α → E}
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
[partial_order α]
{a : α}
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.set_integral_mono_ae_restrict
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ℝ}
{s : set α}
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g s μ)
(h : f ≤ᵐ[μ.restrict s] g) :
theorem
measure_theory.set_integral_mono_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ℝ}
{s : set α}
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g s μ)
(h : f ≤ᵐ[μ] g) :
theorem
measure_theory.set_integral_mono_on
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ℝ}
{s : set α}
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g s μ)
(hs : measurable_set s)
(h : ∀ (x : α), x ∈ s → f x ≤ g x) :
theorem
measure_theory.set_integral_mono_on_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ℝ}
{s : set α}
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g s μ)
(hs : measurable_set s)
(h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x) :
theorem
measure_theory.set_integral_mono
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ℝ}
{s : set α}
(hf : measure_theory.integrable_on f s μ)
(hg : measure_theory.integrable_on g s μ)
(h : f ≤ g) :
theorem
measure_theory.set_integral_ge_of_const_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
{c : ℝ}
(hs : measurable_set s)
(hμs : ⇑μ s ≠ ⊤)
(hf : ∀ (x : α), x ∈ s → c ≤ f x)
(hfint : measure_theory.integrable_on (λ (x : α), f x) s μ) :
theorem
measure_theory.set_integral_nonneg_of_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hf : 0 ≤ᵐ[μ] f) :
theorem
measure_theory.set_integral_nonneg
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : ∀ (a : α), a ∈ s → 0 ≤ f a) :
theorem
measure_theory.set_integral_nonneg_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : ∀ᵐ (a : α) ∂μ, a ∈ s → 0 ≤ f a) :
theorem
measure_theory.set_integral_le_nonneg
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : measure_theory.strongly_measurable f)
(hfi : measure_theory.integrable f μ) :
theorem
measure_theory.set_integral_nonpos_of_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hf : f ≤ᵐ[μ] 0) :
theorem
measure_theory.set_integral_nonpos
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : ∀ (a : α), a ∈ s → f a ≤ 0) :
theorem
measure_theory.set_integral_nonpos_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : ∀ᵐ (a : α) ∂μ, a ∈ s → f a ≤ 0) :
theorem
measure_theory.set_integral_nonpos_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ℝ}
{s : set α}
(hs : measurable_set s)
(hf : measure_theory.strongly_measurable f)
(hfi : measure_theory.integrable f μ) :
theorem
measure_theory.integrable_on_Union_of_summable_integral_norm
{α : Type u_1}
{β : Type u_2}
{E : Type u_3}
[measurable_space α]
{μ : measure_theory.measure α}
[normed_add_comm_group E]
[countable β]
{f : α → E}
{s : β → set α}
(hs : ∀ (b : β), measurable_set (s b))
(hi : ∀ (b : β), measure_theory.integrable_on f (s b) μ)
(h : summable (λ (b : β), ∫ (a : α) in s b, ‖f a‖ ∂μ)) :
theorem
measure_theory.integrable_on_Union_of_summable_norm_restrict
{α : Type u_1}
{β : Type u_2}
{E : Type u_3}
[measurable_space α]
{μ : measure_theory.measure α}
[normed_add_comm_group E]
[countable β]
[topological_space α]
[borel_space α]
[topological_space.metrizable_space α]
[measure_theory.is_locally_finite_measure μ]
{f : C(α, E)}
{s : β → topological_space.compacts α}
(hf : summable (λ (i : β), ‖continuous_map.restrict ↑(s i) f‖ * (⇑μ ↑(s i)).to_real)) :
measure_theory.integrable_on ⇑f (⋃ (i : β), ↑(s i)) μ
If s is a countable family of compact sets, f is a continuous function, and the sequence
‖f.restrict (s i)‖ * μ (s i) is summable, then f is integrable on the union of the s i.
theorem
measure_theory.integrable_of_summable_norm_restrict
{α : Type u_1}
{β : Type u_2}
{E : Type u_3}
[measurable_space α]
{μ : measure_theory.measure α}
[normed_add_comm_group E]
[countable β]
[topological_space α]
[borel_space α]
[topological_space.metrizable_space α]
[measure_theory.is_locally_finite_measure μ]
{f : C(α, E)}
{s : β → topological_space.compacts α}
(hf : summable (λ (i : β), ‖continuous_map.restrict ↑(s i) f‖ * (⇑μ ↑(s i)).to_real))
(hs : (⋃ (i : β), ↑(s i)) = set.univ) :
If s is a countable family of compact sets covering α, f is a continuous function, and
the sequence ‖f.restrict (s i)‖ * μ (s i) is summable, then f is integrable.
theorem
antitone.tendsto_set_integral
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
{μ : measure_theory.measure α}
[normed_add_comm_group E]
[complete_space E]
[normed_space ℝ E]
{s : ℕ → set α}
{f : α → E}
(hsm : ∀ (i : ℕ), measurable_set (s i))
(h_anti : antitone s)
(hfi : measure_theory.integrable_on f (s 0) μ) :
Continuity of the set integral #
We prove that for any set s, the function λ f : α →₁[μ] E, ∫ x in s, f x ∂μ is continuous.
theorem
measure_theory.Lp_to_Lp_restrict_add
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{p : ennreal}
{μ : measure_theory.measure α}
(f g : ↥(measure_theory.Lp E p μ))
(s : set α) :
For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by
(Lp.mem_ℒp f).restrict s).to_Lp f. This map is additive.
theorem
measure_theory.Lp_to_Lp_restrict_smul
{α : Type u_1}
{F : Type u_4}
[measurable_space α]
{𝕜 : Type u_5}
[normed_field 𝕜]
[normed_add_comm_group F]
[normed_space 𝕜 F]
{p : ennreal}
{μ : measure_theory.measure α}
(c : 𝕜)
(f : ↥(measure_theory.Lp F p μ))
(s : set α) :
measure_theory.mem_ℒp.to_Lp ⇑(c • f) _ = c • measure_theory.mem_ℒp.to_Lp ⇑f _
For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by
(Lp.mem_ℒp f).restrict s).to_Lp f. This map commutes with scalar multiplication.
theorem
measure_theory.norm_Lp_to_Lp_restrict_le
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{p : ennreal}
{μ : measure_theory.measure α}
(s : set α)
(f : ↥(measure_theory.Lp E p μ)) :
For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by
(Lp.mem_ℒp f).restrict s).to_Lp f. This map is non-expansive.
noncomputable
def
measure_theory.Lp_to_Lp_restrict_clm
(α : Type u_1)
(F : Type u_4)
[measurable_space α]
(𝕜 : Type u_5)
[normed_field 𝕜]
[normed_add_comm_group F]
[normed_space 𝕜 F]
(μ : measure_theory.measure α)
(p : ennreal)
[hp : fact (1 ≤ p)]
(s : set α) :
↥(measure_theory.Lp F p μ) →L[𝕜] ↥(measure_theory.Lp F p (μ.restrict s))
Continuous linear map sending a function of Lp F p μ to the same function in
Lp F p (μ.restrict s).
Equations
- measure_theory.Lp_to_Lp_restrict_clm α F 𝕜 μ p s = {to_fun := λ (f : ↥(measure_theory.Lp F p μ)), measure_theory.mem_ℒp.to_Lp ⇑f _, map_add' := _, map_smul' := _}.mk_continuous 1 _
theorem
measure_theory.Lp_to_Lp_restrict_clm_coe_fn
{α : Type u_1}
{F : Type u_4}
[measurable_space α]
(𝕜 : Type u_5)
[normed_field 𝕜]
[normed_add_comm_group F]
[normed_space 𝕜 F]
{p : ennreal}
{μ : measure_theory.measure α}
[hp : fact (1 ≤ p)]
(s : set α)
(f : ↥(measure_theory.Lp F p μ)) :
@[continuity]
theorem
measure_theory.continuous_set_integral
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[normed_space ℝ E]
[complete_space E]
(s : set α) :
continuous (λ (f : ↥(measure_theory.Lp E 1 μ)), ∫ (x : α) in s, ⇑f x ∂μ)
theorem
filter.tendsto.integral_sub_linear_is_o_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
{ι : Type u_5}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
{l : filter α}
[l.is_measurably_generated]
{f : α → E}
{b : E}
(h : filter.tendsto f (l ⊓ μ.ae) (nhds b))
(hfm : strongly_measurable_at_filter f l μ)
(hμ : μ.finite_at_filter l)
{s : ι → set α}
{li : filter ι}
(hs : filter.tendsto s li l.small_sets)
(m : ι → ℝ := λ (i : ι), (⇑μ (s i)).to_real)
(hsμ : (λ (i : ι), (⇑μ (s i)).to_real) =ᶠ[li] m . "refl") :
Fundamental theorem of calculus for set integrals: if μ is a measure that is finite at a
filter l and f is a measurable function that has a finite limit b at l ⊓ μ.ae, then ∫ x in s i, f x ∂μ = μ (s i) • b + o(μ (s i)) at a filter li provided that s i tends to l.small_sets
along li. Since μ (s i) is an ℝ≥0∞ number, we use (μ (s i)).to_real in the actual statement.
Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional
argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.
theorem
continuous_within_at.integral_sub_linear_is_o_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
{ι : Type u_5}
[normed_add_comm_group E]
[topological_space α]
[opens_measurable_space α]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a : α}
{t : set α}
{f : α → E}
(ha : continuous_within_at f t a)
(ht : measurable_set t)
(hfm : strongly_measurable_at_filter f (nhds_within a t) μ)
{s : ι → set α}
{li : filter ι}
(hs : filter.tendsto s li (nhds_within a t).small_sets)
(m : ι → ℝ := λ (i : ι), (⇑μ (s i)).to_real)
(hsμ : (λ (i : ι), (⇑μ (s i)).to_real) =ᶠ[li] m . "refl") :
Fundamental theorem of calculus for set integrals, nhds_within version: if μ is a locally
finite measure and f is an almost everywhere measurable function that is continuous at a point a
within a measurable set t, then ∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i)) at a filter li
provided that s i tends to (𝓝[t] a).small_sets along li. Since μ (s i) is an ℝ≥0∞
number, we use (μ (s i)).to_real in the actual statement.
Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional
argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.
theorem
continuous_at.integral_sub_linear_is_o_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
{ι : Type u_5}
[normed_add_comm_group E]
[topological_space α]
[opens_measurable_space α]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a : α}
{f : α → E}
(ha : continuous_at f a)
(hfm : strongly_measurable_at_filter f (nhds a) μ)
{s : ι → set α}
{li : filter ι}
(hs : filter.tendsto s li (nhds a).small_sets)
(m : ι → ℝ := λ (i : ι), (⇑μ (s i)).to_real)
(hsμ : (λ (i : ι), (⇑μ (s i)).to_real) =ᶠ[li] m . "refl") :
Fundamental theorem of calculus for set integrals, nhds version: if μ is a locally finite
measure and f is an almost everywhere measurable function that is continuous at a point a, then
∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i)) at li provided that s tends to
(𝓝 a).small_sets along li. Sinceμ (s i)is anℝ≥0∞number, we use(μ (s i)).to_real` in
the actual statement.
Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional
argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.
theorem
continuous_on.integral_sub_linear_is_o_ae
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
{ι : Type u_5}
[normed_add_comm_group E]
[topological_space α]
[opens_measurable_space α]
[normed_space ℝ E]
[complete_space E]
[second_countable_topology_either α E]
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a : α}
{t : set α}
{f : α → E}
(hft : continuous_on f t)
(ha : a ∈ t)
(ht : measurable_set t)
{s : ι → set α}
{li : filter ι}
(hs : filter.tendsto s li (nhds_within a t).small_sets)
(m : ι → ℝ := λ (i : ι), (⇑μ (s i)).to_real)
(hsμ : (λ (i : ι), (⇑μ (s i)).to_real) =ᶠ[li] m . "refl") :
Fundamental theorem of calculus for set integrals, nhds_within version: if μ is a locally
finite measure, f is continuous on a measurable set t, and a ∈ t, then ∫ x in (s i), f x ∂μ = μ (s i) • f a + o(μ (s i)) at li provided that s i tends to (𝓝[t] a).small_sets along li.
Since μ (s i) is an ℝ≥0∞ number, we use (μ (s i)).to_real in the actual statement.
Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional
argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.
Continuous linear maps composed with integration #
The goal of this section is to prove that integration commutes with continuous linear maps.
This holds for simple functions. The general result follows from the continuity of all involved
operations on the space L¹. Note that composition by a continuous linear map on L¹ is not just
the composition, as we are dealing with classes of functions, but it has already been defined
as continuous_linear_map.comp_Lp. We take advantage of this construction here.
theorem
continuous_linear_map.integral_comp_Lp
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
{p : ennreal}
[complete_space F]
[normed_space ℝ F]
(L : E →L[𝕜] F)
(φ : ↥(measure_theory.Lp E p μ)) :
theorem
continuous_linear_map.set_integral_comp_Lp
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
{p : ennreal}
[complete_space F]
[normed_space ℝ F]
(L : E →L[𝕜] F)
(φ : ↥(measure_theory.Lp E p μ))
{s : set α}
(hs : measurable_set s) :
theorem
continuous_linear_map.continuous_integral_comp_L1
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
(L : E →L[𝕜] F) :
continuous (λ (φ : ↥(measure_theory.Lp E 1 μ)), ∫ (a : α), ⇑L (⇑φ a) ∂μ)
theorem
continuous_linear_map.integral_comp_comm
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
[complete_space E]
[normed_space ℝ E]
(L : E →L[𝕜] F)
{φ : α → E}
(φ_int : measure_theory.integrable φ μ) :
theorem
continuous_linear_map.integral_apply
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[complete_space E]
[normed_space ℝ E]
{H : Type u_2}
[normed_add_comm_group H]
[normed_space 𝕜 H]
{φ : α → (H →L[𝕜] E)}
(φ_int : measure_theory.integrable φ μ)
(v : H) :
theorem
continuous_linear_map.integral_comp_comm'
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
[complete_space E]
[normed_space ℝ E]
(L : E →L[𝕜] F)
{K : nnreal}
(hL : antilipschitz_with K ⇑L)
(φ : α → E) :
theorem
continuous_linear_map.integral_comp_L1_comm
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
[complete_space E]
[normed_space ℝ E]
(L : E →L[𝕜] F)
(φ : ↥(measure_theory.Lp E 1 μ)) :
theorem
linear_isometry.integral_comp_comm
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
[complete_space E]
[normed_space ℝ E]
(L : E →ₗᵢ[𝕜] F)
(φ : α → E) :
theorem
continuous_linear_equiv.integral_comp_comm
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space F]
[normed_space ℝ F]
[complete_space E]
[normed_space ℝ E]
(L : E ≃L[𝕜] F)
(φ : α → E) :
theorem
integral_re
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜}
(hf : measure_theory.integrable f μ) :
theorem
integral_im
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜}
(hf : measure_theory.integrable f μ) :
theorem
integral_conj
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜} :
theorem
integral_coe_re_add_coe_im
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜}
(hf : measure_theory.integrable f μ) :
theorem
integral_re_add_im
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜}
(hf : measure_theory.integrable f μ) :
theorem
set_integral_re_add_im
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{𝕜 : Type u_6}
[is_R_or_C 𝕜]
{f : α → 𝕜}
{i : set α}
(hf : measure_theory.integrable_on f i μ) :
theorem
fst_integral
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[normed_add_comm_group F]
[complete_space E]
[normed_space ℝ E]
[complete_space F]
[normed_space ℝ F]
{f : α → E × F}
(hf : measure_theory.integrable f μ) :
theorem
snd_integral
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[normed_add_comm_group F]
[complete_space E]
[normed_space ℝ E]
[complete_space F]
[normed_space ℝ F]
{f : α → E × F}
(hf : measure_theory.integrable f μ) :
theorem
integral_pair
{α : Type u_1}
{E : Type u_3}
{F : Type u_4}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[normed_add_comm_group F]
[complete_space E]
[normed_space ℝ E]
[complete_space F]
[normed_space ℝ F]
{f : α → E}
{g : α → F}
(hf : measure_theory.integrable f μ)
(hg : measure_theory.integrable g μ) :
theorem
integral_smul_const
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{𝕜 : Type u_2}
[is_R_or_C 𝕜]
[normed_space 𝕜 E]
(f : α → 𝕜)
(c : E) :
theorem
integral_with_density_eq_integral_smul
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → nnreal}
(f_meas : measurable f)
(g : α → E) :
theorem
integral_with_density_eq_integral_smul₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → nnreal}
(hf : ae_measurable f μ)
(g : α → E) :
theorem
set_integral_with_density_eq_set_integral_smul
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → nnreal}
(f_meas : measurable f)
(g : α → E)
{s : set α}
(hs : measurable_set s) :
theorem
set_integral_with_density_eq_set_integral_smul₀
{α : Type u_1}
{E : Type u_3}
[measurable_space α]
[normed_add_comm_group E]
{μ : measure_theory.measure α}
[complete_space E]
[normed_space ℝ E]
{f : α → nnreal}
{s : set α}
(hf : ae_measurable f (μ.restrict s))
(g : α → E)
(hs : measurable_set s) :
theorem
measure_le_lintegral_thickened_indicator_aux
{α : Type u_1}
[measurable_space α]
[pseudo_emetric_space α]
(μ : measure_theory.measure α)
{E : set α}
(E_mble : measurable_set E)
(δ : ℝ) :
theorem
measure_le_lintegral_thickened_indicator
{α : Type u_1}
[measurable_space α]
[pseudo_emetric_space α]
(μ : measure_theory.measure α)
{E : set α}
(E_mble : measurable_set E)
{δ : ℝ}
(δ_pos : 0 < δ) :
theorem
measure_theory.integrable.simple_func_mul
{β : Type u_2}
{f : β → ℝ}
{m0 : measurable_space β}
{μ : measure_theory.measure β}
(g : measure_theory.simple_func β ℝ)
(hf : measure_theory.integrable f μ) :
measure_theory.integrable (⇑g * f) μ
theorem
measure_theory.integrable.simple_func_mul'
{β : Type u_2}
{f : β → ℝ}
{m m0 : measurable_space β}
{μ : measure_theory.measure β}
(hm : m ≤ m0)
(g : measure_theory.simple_func β ℝ)
(hf : measure_theory.integrable f μ) :
measure_theory.integrable (⇑g * f) μ