measure_theory.measure.content ⟷ Mathlib.MeasureTheory.Measure.Content

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -188,7 +188,7 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   conv at this =>
     rhs
     rw [inner_content];
-  simp only [lt_iSup_iff] at this 
+  simp only [lt_iSup_iff] at this
   rcases this with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
@@ -227,7 +227,7 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
   It required for the API of `induced_outer_measure`. -/
 theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
     μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
-  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this 
+  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 -/
 
@@ -427,13 +427,13 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { L : compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
   rw [ENNReal.iSup_add]
-  refine' iSup_le _; rintro ⟨L, hL⟩; simp only [subset_inter_iff] at hL 
+  refine' iSup_le _; rintro ⟨L, hL⟩; simp only [subset_inter_iff] at hL
   have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
   refine' le_trans (add_le_add_left (μ.outer_measure.mono' this) _) _
   rw [μ.outer_measure_of_is_open (↑U' \ L) (IsOpen.sdiff U'.2 L.2.IsClosed)]
   simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { M : compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [ENNReal.add_iSup]; refine' iSup_le _; rintro ⟨M, hM⟩; simp only [subset_diff] at hM 
+  rw [ENNReal.add_iSup]; refine' iSup_le _; rintro ⟨M, hM⟩; simp only [subset_diff] at hM
   have : (↑(L ⊔ M) : Set G) ⊆ U' := by
     simp only [union_subset_iff, compacts.coe_sup, hM, hL, and_self_iff]
   rw [μ.outer_measure_of_is_open (↑U') U'.2]
@@ -462,10 +462,10 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
   have : μ.measure.outer_regular :=
     by
     refine' ⟨fun A hA r (hr : _ < _) => _⟩
-    rw [μ.measure_apply hA, outer_measure_eq_infi] at hr 
-    simp only [iInf_lt_iff] at hr 
+    rw [μ.measure_apply hA, outer_measure_eq_infi] at hr
+    simp only [iInf_lt_iff] at hr
     rcases hr with ⟨U, hUo, hAU, hr⟩
-    rw [← μ.outer_measure_of_is_open U hUo, ← μ.measure_apply hUo.measurable_set] at hr 
+    rw [← μ.outer_measure_of_is_open U hUo, ← μ.measure_apply hUo.measurable_set] at hr
     exact ⟨U, hAU, hUo, hr⟩
   have : is_finite_measure_on_compacts μ.measure :=
     by
@@ -473,8 +473,8 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
     rw [measure_apply _ hK.measurable_set]
     exact μ.outer_measure_lt_top_of_is_compact hK
   refine' ⟨fun U hU r hr => _⟩
-  rw [measure_apply _ hU.measurable_set, μ.outer_measure_of_is_open U hU] at hr 
-  simp only [inner_content, lt_iSup_iff] at hr 
+  rw [measure_apply _ hU.measurable_set, μ.outer_measure_of_is_open U hU] at hr
+  simp only [inner_content, lt_iSup_iff] at hr
   rcases hr with ⟨K, hKU, hr⟩
   refine' ⟨K, hKU, K.2, hr.trans_le _⟩
   exact (μ.le_outer_measure_compacts K).trans (le_to_measure_apply _ _ _)
@@ -500,12 +500,12 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
     ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε :=
   by
   by_contra hc
-  simp only [not_exists, not_and, not_le] at hc 
+  simp only [not_exists, not_and, not_le] at hc
   have lower_bound_infi :
     μ K + ε ≤
       ⨅ (K' : TopologicalSpace.Compacts G) (h : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
     le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
-  rw [← H] at lower_bound_infi 
+  rw [← H] at lower_bound_infi
   exact
     (lt_self_iff_false (μ K)).mp
       (lt_of_le_of_lt' lower_bound_infi

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -236,7 +236,7 @@ theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
   refine' (compacts.equiv f).Surjective.iSup_congr _ fun K => iSup_congr_Prop image_subset_iff _
-  intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, ENNReal.coe_eq_coe, compacts.equiv]
+  intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, ENNReal.coe_inj, compacts.equiv]
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
-import Mathbin.MeasureTheory.Measure.MeasureSpace
-import Mathbin.MeasureTheory.Measure.Regular
-import Mathbin.Topology.Sets.Compacts
+import MeasureTheory.Measure.MeasureSpace
+import MeasureTheory.Measure.Regular
+import Topology.Sets.Compacts
 
 #align_import measure_theory.measure.content from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
-
-! This file was ported from Lean 3 source module measure_theory.measure.content
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Measure.MeasureSpace
 import Mathbin.MeasureTheory.Measure.Regular
 import Mathbin.Topology.Sets.Compacts
 
+#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
 /-!
 # Contents
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -102,26 +102,36 @@ theorem apply_eq_coe_toFun (K : Compacts G) : μ K = μ.toFun K :=
 #align measure_theory.content.apply_eq_coe_to_fun MeasureTheory.Content.apply_eq_coe_toFun
 -/
 
+#print MeasureTheory.Content.mono /-
 theorem mono (K₁ K₂ : Compacts G) (h : (K₁ : Set G) ⊆ K₂) : μ K₁ ≤ μ K₂ := by
   simp [apply_eq_coe_to_fun, μ.mono' _ _ h]
 #align measure_theory.content.mono MeasureTheory.Content.mono
+-/
 
+#print MeasureTheory.Content.sup_disjoint /-
 theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂) :
     μ (K₁ ⊔ K₂) = μ K₁ + μ K₂ := by simp [apply_eq_coe_to_fun, μ.sup_disjoint' _ _ h]
 #align measure_theory.content.sup_disjoint MeasureTheory.Content.sup_disjoint
+-/
 
+#print MeasureTheory.Content.sup_le /-
 theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ := by
   simp only [apply_eq_coe_to_fun]; norm_cast; exact μ.sup_le' _ _
 #align measure_theory.content.sup_le MeasureTheory.Content.sup_le
+-/
 
+#print MeasureTheory.Content.lt_top /-
 theorem lt_top (K : Compacts G) : μ K < ∞ :=
   ENNReal.coe_lt_top
 #align measure_theory.content.lt_top MeasureTheory.Content.lt_top
+-/
 
+#print MeasureTheory.Content.empty /-
 theorem empty : μ ⊥ = 0 := by
   have := μ.sup_disjoint' ⊥ ⊥
   simpa [apply_eq_coe_to_fun] using this
 #align measure_theory.content.empty MeasureTheory.Content.empty
+-/
 
 #print MeasureTheory.Content.innerContent /-
 /-- Constructing the inner content of a content. From a content defined on the compact sets, we
@@ -132,15 +142,19 @@ def innerContent (U : Opens G) : ℝ≥0∞ :=
 #align measure_theory.content.inner_content MeasureTheory.Content.innerContent
 -/
 
+#print MeasureTheory.Content.le_innerContent /-
 theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
     μ K ≤ μ.innerContent U :=
   le_iSup_of_le K <| le_iSup _ h2
 #align measure_theory.content.le_inner_content MeasureTheory.Content.le_innerContent
+-/
 
+#print MeasureTheory.Content.innerContent_le /-
 theorem innerContent_le (U : Opens G) (K : Compacts G) (h2 : (U : Set G) ⊆ K) :
     μ.innerContent U ≤ μ K :=
   iSup₂_le fun K' hK' => μ.mono _ _ (Subset.trans hK' h2)
 #align measure_theory.content.inner_content_le MeasureTheory.Content.innerContent_le
+-/
 
 #print MeasureTheory.Content.innerContent_of_isCompact /-
 theorem innerContent_of_isCompact {K : Set G} (h1K : IsCompact K) (h2K : IsOpen K) :
@@ -158,12 +172,15 @@ theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
 #align measure_theory.content.inner_content_bot MeasureTheory.Content.innerContent_bot
 -/
 
+#print MeasureTheory.Content.innerContent_mono /-
 /-- This is "unbundled", because that it required for the API of `induced_outer_measure`. -/
 theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono MeasureTheory.Content.innerContent_mono
+-/
 
+#print MeasureTheory.Content.innerContent_exists_compact /-
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε :=
   by
@@ -178,7 +195,9 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   rcases this with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
+-/
 
+#print MeasureTheory.Content.innerContent_iSup_nat /-
 /-- The inner content of a supremum of opens is at most the sum of the individual inner
 contents. -/
 theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
@@ -203,7 +222,9 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
   intro i hi; refine' le_trans _ (le_iSup _ (L i))
   refine' le_trans _ (le_iSup _ (h2K' i)); rfl
 #align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_nat
+-/
 
+#print MeasureTheory.Content.innerContent_iUnion_nat /-
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `inner_content_Sup_nat`.
   It required for the API of `induced_outer_measure`. -/
@@ -211,7 +232,9 @@ theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i
     μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
   have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this 
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
+-/
 
+#print MeasureTheory.Content.innerContent_comap /-
 theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
@@ -219,7 +242,9 @@ theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K
   intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, ENNReal.coe_eq_coe, compacts.equiv]
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
+-/
 
+#print MeasureTheory.Content.is_mul_left_invariant_innerContent /-
 @[to_additive]
 theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
@@ -228,7 +253,9 @@ theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
   convert μ.inner_content_comap (Homeomorph.mulLeft g) (fun K => h g) U
 #align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mul_left_invariant_innerContent
 #align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_innerContent
+-/
 
+#print MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant /-
 @[to_additive]
 theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -246,11 +273,14 @@ theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [Topolog
   simp only [μ.is_mul_left_invariant_inner_content h3, Finset.sum_const, nsmul_eq_mul, le_refl]
 #align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
 #align measure_theory.content.inner_content_pos_of_is_add_left_invariant MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant
+-/
 
+#print MeasureTheory.Content.innerContent_mono' /-
 theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono' MeasureTheory.Content.innerContent_mono'
+-/
 
 section OuterMeasure
 
@@ -276,11 +306,14 @@ theorem outerMeasure_of_isOpen (U : Set G) (hU : IsOpen U) :
 #align measure_theory.content.outer_measure_of_is_open MeasureTheory.Content.outerMeasure_of_isOpen
 -/
 
+#print MeasureTheory.Content.outerMeasure_le /-
 theorem outerMeasure_le (U : Opens G) (K : Compacts G) (hUK : (U : Set G) ⊆ K) :
     μ.OuterMeasure U ≤ μ K :=
   (μ.outerMeasure_opens U).le.trans <| μ.innerContent_le U K hUK
 #align measure_theory.content.outer_measure_le MeasureTheory.Content.outerMeasure_le
+-/
 
+#print MeasureTheory.Content.le_outerMeasure_compacts /-
 theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :=
   by
   rw [content.outer_measure, induced_outer_measure_eq_infi]
@@ -288,16 +321,22 @@ theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :
   · exact μ.inner_content_Union_nat
   · exact μ.inner_content_mono
 #align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compacts
+-/
 
+#print MeasureTheory.Content.outerMeasure_eq_iInf /-
 theorem outerMeasure_eq_iInf (A : Set G) :
     μ.OuterMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (h : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
   inducedOuterMeasure_eq_iInf _ μ.innerContent_iUnion_nat μ.innerContent_mono A
 #align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInf
+-/
 
+#print MeasureTheory.Content.outerMeasure_interior_compacts /-
 theorem outerMeasure_interior_compacts (K : Compacts G) : μ.OuterMeasure (interior K) ≤ μ K :=
   (μ.outerMeasure_opens <| Opens.interior K).le.trans <| μ.innerContent_le _ _ interior_subset
 #align measure_theory.content.outer_measure_interior_compacts MeasureTheory.Content.outerMeasure_interior_compacts
+-/
 
+#print MeasureTheory.Content.outerMeasure_exists_compact /-
 theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure K + ε :=
   by
@@ -305,7 +344,9 @@ theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ 
   rcases μ.inner_content_exists_compact hU hε with ⟨K, h1K, h2K⟩
   exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outer_measure_compacts K) _⟩
 #align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compact
+-/
 
+#print MeasureTheory.Content.outerMeasure_exists_open /-
 theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {ε : ℝ≥0} (hε : ε ≠ 0) :
     ∃ U : Opens G, A ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure A + ε :=
   by
@@ -314,7 +355,9 @@ theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {
     ⟨U, hU, h2U, h3U⟩
   exact ⟨⟨U, hU⟩, h2U, h3U⟩; swap; exact μ.inner_content_Union_nat
 #align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_open
+-/
 
+#print MeasureTheory.Content.outerMeasure_preimage /-
 theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (A : Set G) : μ.OuterMeasure (f ⁻¹' A) = μ.OuterMeasure A :=
   by
@@ -323,7 +366,9 @@ theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ
       (fun s => f.is_open_preimage) _
   intro s hs; convert μ.inner_content_comap f h ⟨s, hs⟩
 #align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimage
+-/
 
+#print MeasureTheory.Content.outerMeasure_lt_top_of_isCompact /-
 theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK : IsCompact K) :
     μ.OuterMeasure K < ∞ :=
   by
@@ -334,7 +379,9 @@ theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK
       apply μ.outer_measure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
     _ < ⊤ := μ.lt_top _
 #align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompact
+-/
 
+#print MeasureTheory.Content.is_mul_left_invariant_outerMeasure /-
 @[to_additive]
 theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
@@ -342,7 +389,9 @@ theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
   convert μ.outer_measure_preimage (Homeomorph.mulLeft g) (fun K => h g) A
 #align measure_theory.content.is_mul_left_invariant_outer_measure MeasureTheory.Content.is_mul_left_invariant_outerMeasure
 #align measure_theory.content.is_add_left_invariant_outer_measure MeasureTheory.Content.is_add_left_invariant_outerMeasure
+-/
 
+#print MeasureTheory.Content.outerMeasure_caratheodory /-
 theorem outerMeasure_caratheodory (A : Set G) :
     measurable_set[μ.OuterMeasure.caratheodory] A ↔
       ∀ U : Opens G, μ.OuterMeasure (U ∩ A) + μ.OuterMeasure (U \ A) ≤ μ.OuterMeasure U :=
@@ -352,7 +401,9 @@ theorem outerMeasure_caratheodory (A : Set G) :
   apply inner_content_Union_nat
   apply inner_content_mono'
 #align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodory
+-/
 
+#print MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant /-
 @[to_additive]
 theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -362,11 +413,10 @@ theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
   exact μ.outer_measure_opens ⟨U, h1U⟩
 #align measure_theory.content.outer_measure_pos_of_is_mul_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant
 #align measure_theory.content.outer_measure_pos_of_is_add_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_add_left_invariant
+-/
 
 variable [S : MeasurableSpace G] [BorelSpace G]
 
-include S
-
 #print MeasureTheory.Content.borel_le_caratheodory /-
 /-- For the outer measure coming from a content, all Borel sets are measurable. -/
 theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
@@ -447,6 +497,7 @@ def ContentRegular :=
 #align measure_theory.content.content_regular MeasureTheory.Content.ContentRegular
 -/
 
+#print MeasureTheory.Content.contentRegular_exists_compact /-
 theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
     {ε : NNReal} (hε : ε ≠ 0) :
     ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε :=
@@ -463,6 +514,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
       (lt_of_le_of_lt' lower_bound_infi
         (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ennreal.coe_ne_zero.mpr hε)))
 #align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compact
+-/
 
 variable [MeasurableSpace G] [T2Space G] [BorelSpace G]
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -333,7 +333,6 @@ theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK
     _ ≤ μ ⟨F, h1F⟩ := by
       apply μ.outer_measure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
     _ < ⊤ := μ.lt_top _
-    
 #align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompact
 
 @[to_additive]
@@ -481,7 +480,6 @@ theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular
       μ.measure ↑K ≤ μ.measure (interior ↑K') := _
       _ ≤ μ K' := _
       _ ≤ μ K + ε := K'_hyp.right
-      
     · rw [μ.measure_apply isOpen_interior.MeasurableSet,
         μ.measure_apply K.is_compact.measurable_set]
       exact μ.outer_measure.mono K'_hyp.left

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -174,7 +174,7 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   conv at this =>
     rhs
     rw [inner_content];
-  simp only [lt_iSup_iff] at this
+  simp only [lt_iSup_iff] at this 
   rcases this with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
@@ -209,7 +209,7 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
   It required for the API of `induced_outer_measure`. -/
 theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
     μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
-  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this
+  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this 
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 
 theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
@@ -301,7 +301,7 @@ theorem outerMeasure_interior_compacts (K : Compacts G) : μ.OuterMeasure (inter
 theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure K + ε :=
   by
-  rw [μ.outer_measure_opens] at hU⊢
+  rw [μ.outer_measure_opens] at hU ⊢
   rcases μ.inner_content_exists_compact hU hε with ⟨K, h1K, h2K⟩
   exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outer_measure_compacts K) _⟩
 #align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compact
@@ -381,13 +381,13 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { L : compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
   rw [ENNReal.iSup_add]
-  refine' iSup_le _; rintro ⟨L, hL⟩; simp only [subset_inter_iff] at hL
+  refine' iSup_le _; rintro ⟨L, hL⟩; simp only [subset_inter_iff] at hL 
   have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
   refine' le_trans (add_le_add_left (μ.outer_measure.mono' this) _) _
   rw [μ.outer_measure_of_is_open (↑U' \ L) (IsOpen.sdiff U'.2 L.2.IsClosed)]
   simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { M : compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [ENNReal.add_iSup]; refine' iSup_le _; rintro ⟨M, hM⟩; simp only [subset_diff] at hM
+  rw [ENNReal.add_iSup]; refine' iSup_le _; rintro ⟨M, hM⟩; simp only [subset_diff] at hM 
   have : (↑(L ⊔ M) : Set G) ⊆ U' := by
     simp only [union_subset_iff, compacts.coe_sup, hM, hL, and_self_iff]
   rw [μ.outer_measure_of_is_open (↑U') U'.2]
@@ -416,10 +416,10 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
   have : μ.measure.outer_regular :=
     by
     refine' ⟨fun A hA r (hr : _ < _) => _⟩
-    rw [μ.measure_apply hA, outer_measure_eq_infi] at hr
-    simp only [iInf_lt_iff] at hr
+    rw [μ.measure_apply hA, outer_measure_eq_infi] at hr 
+    simp only [iInf_lt_iff] at hr 
     rcases hr with ⟨U, hUo, hAU, hr⟩
-    rw [← μ.outer_measure_of_is_open U hUo, ← μ.measure_apply hUo.measurable_set] at hr
+    rw [← μ.outer_measure_of_is_open U hUo, ← μ.measure_apply hUo.measurable_set] at hr 
     exact ⟨U, hAU, hUo, hr⟩
   have : is_finite_measure_on_compacts μ.measure :=
     by
@@ -427,8 +427,8 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
     rw [measure_apply _ hK.measurable_set]
     exact μ.outer_measure_lt_top_of_is_compact hK
   refine' ⟨fun U hU r hr => _⟩
-  rw [measure_apply _ hU.measurable_set, μ.outer_measure_of_is_open U hU] at hr
-  simp only [inner_content, lt_iSup_iff] at hr
+  rw [measure_apply _ hU.measurable_set, μ.outer_measure_of_is_open U hU] at hr 
+  simp only [inner_content, lt_iSup_iff] at hr 
   rcases hr with ⟨K, hKU, hr⟩
   refine' ⟨K, hKU, K.2, hr.trans_le _⟩
   exact (μ.le_outer_measure_compacts K).trans (le_to_measure_apply _ _ _)
@@ -453,12 +453,12 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
     ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε :=
   by
   by_contra hc
-  simp only [not_exists, not_and, not_le] at hc
+  simp only [not_exists, not_and, not_le] at hc 
   have lower_bound_infi :
     μ K + ε ≤
       ⨅ (K' : TopologicalSpace.Compacts G) (h : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
     le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
-  rw [← H] at lower_bound_infi
+  rw [← H] at lower_bound_infi 
   exact
     (lt_self_iff_false (μ K)).mp
       (lt_of_le_of_lt' lower_bound_infi

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -62,7 +62,7 @@ noncomputable section
 
 open Set TopologicalSpace
 
-open NNReal ENNReal MeasureTheory
+open scoped NNReal ENNReal MeasureTheory
 
 namespace MeasureTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -102,52 +102,22 @@ theorem apply_eq_coe_toFun (K : Compacts G) : μ K = μ.toFun K :=
 #align measure_theory.content.apply_eq_coe_to_fun MeasureTheory.Content.apply_eq_coe_toFun
 -/
 
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 theorem mono (K₁ K₂ : Compacts G) (h : (K₁ : Set G) ⊆ K₂) : μ K₁ ≤ μ K₂ := by
   simp [apply_eq_coe_to_fun, μ.mono' _ _ h]
 #align measure_theory.content.mono MeasureTheory.Content.mono
 
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 theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂) :
     μ (K₁ ⊔ K₂) = μ K₁ + μ K₂ := by simp [apply_eq_coe_to_fun, μ.sup_disjoint' _ _ h]
 #align measure_theory.content.sup_disjoint MeasureTheory.Content.sup_disjoint
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.sup_le MeasureTheory.Content.sup_leₓ'. -/
 theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ := by
   simp only [apply_eq_coe_to_fun]; norm_cast; exact μ.sup_le' _ _
 #align measure_theory.content.sup_le MeasureTheory.Content.sup_le
 
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 theorem lt_top (K : Compacts G) : μ K < ∞ :=
   ENNReal.coe_lt_top
 #align measure_theory.content.lt_top MeasureTheory.Content.lt_top
 
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 theorem empty : μ ⊥ = 0 := by
   have := μ.sup_disjoint' ⊥ ⊥
   simpa [apply_eq_coe_to_fun] using this
@@ -162,23 +132,11 @@ def innerContent (U : Opens G) : ℝ≥0∞ :=
 #align measure_theory.content.inner_content MeasureTheory.Content.innerContent
 -/
 
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 theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
     μ K ≤ μ.innerContent U :=
   le_iSup_of_le K <| le_iSup _ h2
 #align measure_theory.content.le_inner_content MeasureTheory.Content.le_innerContent
 
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 theorem innerContent_le (U : Opens G) (K : Compacts G) (h2 : (U : Set G) ⊆ K) :
     μ.innerContent U ≤ μ K :=
   iSup₂_le fun K' hK' => μ.mono _ _ (Subset.trans hK' h2)
@@ -200,24 +158,12 @@ theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
 #align measure_theory.content.inner_content_bot MeasureTheory.Content.innerContent_bot
 -/
 
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 /-- This is "unbundled", because that it required for the API of `induced_outer_measure`. -/
 theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono MeasureTheory.Content.innerContent_mono
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compactₓ'. -/
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε :=
   by
@@ -233,12 +179,6 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_natₓ'. -/
 /-- The inner content of a supremum of opens is at most the sum of the individual inner
 contents. -/
 theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
@@ -264,12 +204,6 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
   refine' le_trans _ (le_iSup _ (h2K' i)); rfl
 #align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_nat
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_natₓ'. -/
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `inner_content_Sup_nat`.
   It required for the API of `induced_outer_measure`. -/
@@ -278,12 +212,6 @@ theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i
   have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comapₓ'. -/
 theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
@@ -292,12 +220,6 @@ theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mul_left_invariant_innerContentₓ'. -/
 @[to_additive]
 theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
@@ -307,12 +229,6 @@ theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
 #align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mul_left_invariant_innerContent
 #align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_innerContent
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariantₓ'. -/
 @[to_additive]
 theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -331,12 +247,6 @@ theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [Topolog
 #align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
 #align measure_theory.content.inner_content_pos_of_is_add_left_invariant MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_mono' MeasureTheory.Content.innerContent_mono'ₓ'. -/
 theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
@@ -366,23 +276,11 @@ theorem outerMeasure_of_isOpen (U : Set G) (hU : IsOpen U) :
 #align measure_theory.content.outer_measure_of_is_open MeasureTheory.Content.outerMeasure_of_isOpen
 -/
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_le MeasureTheory.Content.outerMeasure_leₓ'. -/
 theorem outerMeasure_le (U : Opens G) (K : Compacts G) (hUK : (U : Set G) ⊆ K) :
     μ.OuterMeasure U ≤ μ K :=
   (μ.outerMeasure_opens U).le.trans <| μ.innerContent_le U K hUK
 #align measure_theory.content.outer_measure_le MeasureTheory.Content.outerMeasure_le
 
-/- warning: measure_theory.content.le_outer_measure_compacts -> MeasureTheory.Content.le_outerMeasure_compacts is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compactsₓ'. -/
 theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :=
   by
   rw [content.outer_measure, induced_outer_measure_eq_infi]
@@ -391,33 +289,15 @@ theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :
   · exact μ.inner_content_mono
 #align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compacts
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInfₓ'. -/
 theorem outerMeasure_eq_iInf (A : Set G) :
     μ.OuterMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (h : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
   inducedOuterMeasure_eq_iInf _ μ.innerContent_iUnion_nat μ.innerContent_mono A
 #align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInf
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_interior_compacts MeasureTheory.Content.outerMeasure_interior_compactsₓ'. -/
 theorem outerMeasure_interior_compacts (K : Compacts G) : μ.OuterMeasure (interior K) ≤ μ K :=
   (μ.outerMeasure_opens <| Opens.interior K).le.trans <| μ.innerContent_le _ _ interior_subset
 #align measure_theory.content.outer_measure_interior_compacts MeasureTheory.Content.outerMeasure_interior_compacts
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compactₓ'. -/
 theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure K + ε :=
   by
@@ -426,12 +306,6 @@ theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ 
   exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outer_measure_compacts K) _⟩
 #align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compact
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_openₓ'. -/
 theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {ε : ℝ≥0} (hε : ε ≠ 0) :
     ∃ U : Opens G, A ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure A + ε :=
   by
@@ -441,12 +315,6 @@ theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {
   exact ⟨⟨U, hU⟩, h2U, h3U⟩; swap; exact μ.inner_content_Union_nat
 #align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_open
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimageₓ'. -/
 theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (A : Set G) : μ.OuterMeasure (f ⁻¹' A) = μ.OuterMeasure A :=
   by
@@ -456,12 +324,6 @@ theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ
   intro s hs; convert μ.inner_content_comap f h ⟨s, hs⟩
 #align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimage
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompactₓ'. -/
 theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK : IsCompact K) :
     μ.OuterMeasure K < ∞ :=
   by
@@ -474,12 +336,6 @@ theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK
     
 #align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompact
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.is_mul_left_invariant_outer_measure MeasureTheory.Content.is_mul_left_invariant_outerMeasureₓ'. -/
 @[to_additive]
 theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
@@ -488,12 +344,6 @@ theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
 #align measure_theory.content.is_mul_left_invariant_outer_measure MeasureTheory.Content.is_mul_left_invariant_outerMeasure
 #align measure_theory.content.is_add_left_invariant_outer_measure MeasureTheory.Content.is_add_left_invariant_outerMeasure
 
-/- warning: measure_theory.content.outer_measure_caratheodory -> MeasureTheory.Content.outerMeasure_caratheodory is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodoryₓ'. -/
 theorem outerMeasure_caratheodory (A : Set G) :
     measurable_set[μ.OuterMeasure.caratheodory] A ↔
       ∀ U : Opens G, μ.OuterMeasure (U ∩ A) + μ.OuterMeasure (U \ A) ≤ μ.OuterMeasure U :=
@@ -504,12 +354,6 @@ theorem outerMeasure_caratheodory (A : Set G) :
   apply inner_content_mono'
 #align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodory
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_pos_of_is_mul_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariantₓ'. -/
 @[to_additive]
 theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -604,12 +448,6 @@ def ContentRegular :=
 #align measure_theory.content.content_regular MeasureTheory.Content.ContentRegular
 -/
 
-/- warning: measure_theory.content.content_regular_exists_compact -> MeasureTheory.Content.contentRegular_exists_compact is a dubious translation:
-lean 3 declaration is
-  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), (MeasureTheory.Content.ContentRegular.{u1} G _inst_1 μ) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1) {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K' : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K) (interior.{u1} G _inst_1 (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K'))) (LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K') (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) ((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))) ε))))))
-but is expected to have type
-  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), (MeasureTheory.Content.ContentRegular.{u1} G _inst_1 μ) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1) {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K' : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K) (interior.{u1} G _inst_1 (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K'))) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K') (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (ENNReal.some ε))))))
-Case conversion may be inaccurate. Consider using '#align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compactₓ'. -/
 theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
     {ε : NNReal} (hε : ε ≠ 0) :
     ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -128,11 +128,8 @@ lean 3 declaration is
 but is expected to have type
   forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (Sup.sup.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.instSupCompacts.{u1} G _inst_1) K₁ K₂)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₁) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₂))
 Case conversion may be inaccurate. Consider using '#align measure_theory.content.sup_le MeasureTheory.Content.sup_leₓ'. -/
-theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ :=
-  by
-  simp only [apply_eq_coe_to_fun]
-  norm_cast
-  exact μ.sup_le' _ _
+theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ := by
+  simp only [apply_eq_coe_to_fun]; norm_cast; exact μ.sup_le' _ _
 #align measure_theory.content.sup_le MeasureTheory.Content.sup_le
 
 /- warning: measure_theory.content.lt_top -> MeasureTheory.Content.lt_top is a dubious translation:
@@ -197,14 +194,9 @@ theorem innerContent_of_isCompact {K : Set G} (h1K : IsCompact K) (h2K : IsOpen
 #print MeasureTheory.Content.innerContent_bot /-
 theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
   by
-  refine' le_antisymm _ (zero_le _)
-  rw [← μ.empty]
+  refine' le_antisymm _ (zero_le _); rw [← μ.empty]
   refine' iSup₂_le fun K hK => _
-  have : K = ⊥ := by
-    ext1
-    rw [subset_empty_iff.mp hK, compacts.coe_bot]
-  rw [this]
-  rfl
+  have : K = ⊥ := by ext1; rw [subset_empty_iff.mp hK, compacts.coe_bot]; rw [this]; rfl
 #align measure_theory.content.inner_content_bot MeasureTheory.Content.innerContent_bot
 -/
 
@@ -254,29 +246,22 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
   by
   have h3 : ∀ (t : Finset ℕ) (K : ℕ → compacts G), μ (t.sup K) ≤ t.Sum fun i => μ (K i) :=
     by
-    intro t K
-    refine' Finset.induction_on t _ _
+    intro t K; refine' Finset.induction_on t _ _
     · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
-    · intro n s hn ih
-      rw [Finset.sup_insert, Finset.sum_insert hn]
+    · intro n s hn ih; rw [Finset.sup_insert, Finset.sum_insert hn]
       exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
   refine' iSup₂_le fun K hK => _
-  obtain ⟨t, ht⟩ := K.is_compact.elim_finite_subcover _ (fun i => (U i).IsOpen) _
-  swap
+  obtain ⟨t, ht⟩ := K.is_compact.elim_finite_subcover _ (fun i => (U i).IsOpen) _; swap
   · rwa [← opens.coe_supr]
   rcases K.is_compact.finite_compact_cover t (coe ∘ U) (fun i _ => (U _).IsOpen)
       (by simp only [ht]) with
     ⟨K', h1K', h2K', h3K'⟩
   let L : ℕ → compacts G := fun n => ⟨K' n, h1K' n⟩
   convert le_trans (h3 t L) _
-  · ext1
-    rw [compacts.coe_finset_sup, Finset.sup_eq_iSup]
-    exact h3K'
+  · ext1; rw [compacts.coe_finset_sup, Finset.sup_eq_iSup]; exact h3K'
   refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)
-  intro i hi
-  refine' le_trans _ (le_iSup _ (L i))
-  refine' le_trans _ (le_iSup _ (h2K' i))
-  rfl
+  intro i hi; refine' le_trans _ (le_iSup _ (L i))
+  refine' le_trans _ (le_iSup _ (h2K' i)); rfl
 #align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_nat
 
 /- warning: measure_theory.content.inner_content_Union_nat -> MeasureTheory.Content.innerContent_iUnion_nat is a dubious translation:
@@ -289,10 +274,8 @@ Case conversion may be inaccurate. Consider using '#align measure_theory.content
   This is the "unbundled" version of `inner_content_Sup_nat`.
   It required for the API of `induced_outer_measure`. -/
 theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
-    μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ :=
-  by
-  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩
-  rwa [opens.supr_def] at this
+    μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
+  have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩; rwa [opens.supr_def] at this
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 
 /- warning: measure_theory.content.inner_content_comap -> MeasureTheory.Content.innerContent_comap is a dubious translation:
@@ -335,8 +318,7 @@ theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [Topolog
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
     (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U :=
   by
-  have : (interior (U : Set G)).Nonempty
-  rwa [U.is_open.interior_eq]
+  have : (interior (U : Set G)).Nonempty; rwa [U.is_open.interior_eq]
   rcases compact_covered_by_mul_left_translates K.2 this with ⟨s, hs⟩
   suffices μ K ≤ s.card * μ.inner_content U by
     exact (ennreal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2
@@ -552,23 +534,16 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   rw [μ.outer_measure_caratheodory]
   intro U'
   rw [μ.outer_measure_of_is_open ((U' : Set G) ∩ U) (U'.is_open.inter hU)]
-  simp only [inner_content, iSup_subtype']
-  rw [opens.coe_mk]
+  simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { L : compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
   rw [ENNReal.iSup_add]
-  refine' iSup_le _
-  rintro ⟨L, hL⟩
-  simp only [subset_inter_iff] at hL
+  refine' iSup_le _; rintro ⟨L, hL⟩; simp only [subset_inter_iff] at hL
   have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
   refine' le_trans (add_le_add_left (μ.outer_measure.mono' this) _) _
   rw [μ.outer_measure_of_is_open (↑U' \ L) (IsOpen.sdiff U'.2 L.2.IsClosed)]
-  simp only [inner_content, iSup_subtype']
-  rw [opens.coe_mk]
+  simp only [inner_content, iSup_subtype']; rw [opens.coe_mk]
   haveI : Nonempty { M : compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [ENNReal.add_iSup]
-  refine' iSup_le _
-  rintro ⟨M, hM⟩
-  simp only [subset_diff] at hM
+  rw [ENNReal.add_iSup]; refine' iSup_le _; rintro ⟨M, hM⟩; simp only [subset_diff] at hM
   have : (↑(L ⊔ M) : Set G) ⊆ U' := by
     simp only [union_subset_iff, compacts.coe_sup, hM, hL, and_self_iff]
   rw [μ.outer_measure_of_is_open (↑U') U'.2]

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 
 ! This file was ported from Lean 3 source module measure_theory.measure.content
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Topology.Sets.Compacts
 /-!
 # Contents
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we work with *contents*. A content `λ` is a function from a certain class of subsets
 (such as the compact subsets) to `ℝ≥0` that is
 * additive: If `K₁` and `K₂` are disjoint sets in the domain of `λ`,

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -65,6 +65,7 @@ namespace MeasureTheory
 
 variable {G : Type w} [TopologicalSpace G]
 
+#print MeasureTheory.Content /-
 /-- A content is an additive function on compact sets taking values in `ℝ≥0`. It is a device
 from which one can define a measure. -/
 structure Content (G : Type w) [TopologicalSpace G] where
@@ -74,6 +75,7 @@ structure Content (G : Type w) [TopologicalSpace G] where
     ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → to_fun (K₁ ⊔ K₂) = to_fun K₁ + to_fun K₂
   sup_le' : ∀ K₁ K₂ : Compacts G, to_fun (K₁ ⊔ K₂) ≤ to_fun K₁ + to_fun K₂
 #align measure_theory.content MeasureTheory.Content
+-/
 
 instance : Inhabited (Content G) :=
   ⟨{  toFun := fun K => 0
@@ -91,18 +93,38 @@ namespace Content
 
 variable (μ : Content G)
 
+#print MeasureTheory.Content.apply_eq_coe_toFun /-
 theorem apply_eq_coe_toFun (K : Compacts G) : μ K = μ.toFun K :=
   rfl
 #align measure_theory.content.apply_eq_coe_to_fun MeasureTheory.Content.apply_eq_coe_toFun
+-/
 
+/- warning: measure_theory.content.mono -> MeasureTheory.Content.mono is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K₁) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K₂)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₁) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₂))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K₁) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K₂)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₁) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₂))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.mono MeasureTheory.Content.monoₓ'. -/
 theorem mono (K₁ K₂ : Compacts G) (h : (K₁ : Set G) ⊆ K₂) : μ K₁ ≤ μ K₂ := by
   simp [apply_eq_coe_to_fun, μ.mono' _ _ h]
 #align measure_theory.content.mono MeasureTheory.Content.mono
 
+/- warning: measure_theory.content.sup_disjoint -> MeasureTheory.Content.sup_disjoint is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), (Disjoint.{u1} (Set.{u1} G) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} G) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} G) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} G) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} G) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} G) (Set.completeBooleanAlgebra.{u1} G)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} G) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} G) (Set.booleanAlgebra.{u1} G))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K₁) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K₂)) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (Sup.sup.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.hasSup.{u1} G _inst_1) K₁ K₂)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₁) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₂)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), (Disjoint.{u1} (Set.{u1} G) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} G) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} G) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} G) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} G) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} G) (Set.instCompleteBooleanAlgebraSet.{u1} G)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} G) (Preorder.toLE.{u1} (Set.{u1} G) (PartialOrder.toPreorder.{u1} (Set.{u1} G) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} G) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} G) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} G) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} G) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} G) (Set.instCompleteBooleanAlgebraSet.{u1} G)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} G) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} G) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} G) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} G) (Set.instCompleteBooleanAlgebraSet.{u1} G)))))) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K₁) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K₂)) -> (Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (Sup.sup.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.instSupCompacts.{u1} G _inst_1) K₁ K₂)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₁) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₂)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.sup_disjoint MeasureTheory.Content.sup_disjointₓ'. -/
 theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂) :
     μ (K₁ ⊔ K₂) = μ K₁ + μ K₂ := by simp [apply_eq_coe_to_fun, μ.sup_disjoint' _ _ h]
 #align measure_theory.content.sup_disjoint MeasureTheory.Content.sup_disjoint
 
+/- warning: measure_theory.content.sup_le -> MeasureTheory.Content.sup_le is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (Sup.sup.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.hasSup.{u1} G _inst_1) K₁ K₂)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₁) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K₂))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K₁ : TopologicalSpace.Compacts.{u1} G _inst_1) (K₂ : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (Sup.sup.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.instSupCompacts.{u1} G _inst_1) K₁ K₂)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₁) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K₂))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.sup_le MeasureTheory.Content.sup_leₓ'. -/
 theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂ :=
   by
   simp only [apply_eq_coe_to_fun]
@@ -110,37 +132,66 @@ theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K
   exact μ.sup_le' _ _
 #align measure_theory.content.sup_le MeasureTheory.Content.sup_le
 
+/- warning: measure_theory.content.lt_top -> MeasureTheory.Content.lt_top is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), 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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), 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))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.lt_top MeasureTheory.Content.lt_topₓ'. -/
 theorem lt_top (K : Compacts G) : μ K < ∞ :=
   ENNReal.coe_lt_top
 #align measure_theory.content.lt_top MeasureTheory.Content.lt_top
 
+/- warning: measure_theory.content.empty -> MeasureTheory.Content.empty is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (Bot.bot.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.hasBot.{u1} G _inst_1))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (Bot.bot.{u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (TopologicalSpace.Compacts.instBotCompacts.{u1} G _inst_1))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.empty MeasureTheory.Content.emptyₓ'. -/
 theorem empty : μ ⊥ = 0 := by
   have := μ.sup_disjoint' ⊥ ⊥
   simpa [apply_eq_coe_to_fun] using this
 #align measure_theory.content.empty MeasureTheory.Content.empty
 
+#print MeasureTheory.Content.innerContent /-
 /-- Constructing the inner content of a content. From a content defined on the compact sets, we
   obtain a function defined on all open sets, by taking the supremum of the content of all compact
   subsets. -/
 def innerContent (U : Opens G) : ℝ≥0∞ :=
   ⨆ (K : Compacts G) (h : (K : Set G) ⊆ U), μ K
 #align measure_theory.content.inner_content MeasureTheory.Content.innerContent
+-/
 
+/- warning: measure_theory.content.le_inner_content -> MeasureTheory.Content.le_innerContent is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1) (U : TopologicalSpace.Opens.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1) (U : TopologicalSpace.Opens.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.le_inner_content MeasureTheory.Content.le_innerContentₓ'. -/
 theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
     μ K ≤ μ.innerContent U :=
   le_iSup_of_le K <| le_iSup _ h2
 #align measure_theory.content.le_inner_content MeasureTheory.Content.le_innerContent
 
+/- warning: measure_theory.content.inner_content_le -> MeasureTheory.Content.innerContent_le is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (U : TopologicalSpace.Opens.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (U : TopologicalSpace.Opens.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_le MeasureTheory.Content.innerContent_leₓ'. -/
 theorem innerContent_le (U : Opens G) (K : Compacts G) (h2 : (U : Set G) ⊆ K) :
     μ.innerContent U ≤ μ K :=
   iSup₂_le fun K' hK' => μ.mono _ _ (Subset.trans hK' h2)
 #align measure_theory.content.inner_content_le MeasureTheory.Content.innerContent_le
 
+#print MeasureTheory.Content.innerContent_of_isCompact /-
 theorem innerContent_of_isCompact {K : Set G} (h1K : IsCompact K) (h2K : IsOpen K) :
     μ.innerContent ⟨K, h2K⟩ = μ ⟨K, h1K⟩ :=
   le_antisymm (iSup₂_le fun K' hK' => μ.mono _ ⟨K, h1K⟩ hK') (μ.le_innerContent _ _ Subset.rfl)
 #align measure_theory.content.inner_content_of_is_compact MeasureTheory.Content.innerContent_of_isCompact
+-/
 
+#print MeasureTheory.Content.innerContent_bot /-
 theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
   by
   refine' le_antisymm _ (zero_le _)
@@ -152,13 +203,26 @@ theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
   rw [this]
   rfl
 #align measure_theory.content.inner_content_bot MeasureTheory.Content.innerContent_bot
+-/
 
+/- warning: measure_theory.content.inner_content_mono -> MeasureTheory.Content.innerContent_mono is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {{U : Set.{u1} G}} {{V : Set.{u1} G}} (hU : IsOpen.{u1} G _inst_1 U) (hV : IsOpen.{u1} G _inst_1 V), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) U V) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 V hV)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {{U : Set.{u1} G}} {{V : Set.{u1} G}} (hU : IsOpen.{u1} G _inst_1 U) (hV : IsOpen.{u1} G _inst_1 V), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) U V) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 V hV)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_mono MeasureTheory.Content.innerContent_monoₓ'. -/
 /-- This is "unbundled", because that it required for the API of `induced_outer_measure`. -/
 theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono MeasureTheory.Content.innerContent_mono
 
+/- warning: measure_theory.content.inner_content_exists_compact -> MeasureTheory.Content.innerContent_exists_compact is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {U : TopologicalSpace.Opens.{u1} G _inst_1}, (Ne.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) ((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))) ε))))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {U : TopologicalSpace.Opens.{u1} G _inst_1}, (Ne.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (ENNReal.some ε))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compactₓ'. -/
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε :=
   by
@@ -174,9 +238,15 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
 
+/- warning: measure_theory.content.inner_content_Sup_nat -> MeasureTheory.Content.innerContent_iSup_nat is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (U : Nat -> (TopologicalSpace.Opens.{u1} G _inst_1)), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (iSup.{u1, 1} (TopologicalSpace.Opens.{u1} G _inst_1) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1))) Nat (fun (i : Nat) => U i))) (tsum.{0, 0} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace Nat (fun (i : Nat) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (U i)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (U : Nat -> (TopologicalSpace.Opens.{u1} G _inst_1)), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (iSup.{u1, 1} (TopologicalSpace.Opens.{u1} G _inst_1) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1))) Nat (fun (i : Nat) => U i))) (tsum.{0, 0} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal Nat (fun (i : Nat) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (U i)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_natₓ'. -/
 /-- The inner content of a supremum of opens is at most the sum of the individual inner
 contents. -/
-theorem innerContent_Sup_nat [T2Space G] (U : ℕ → Opens G) :
+theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
     μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) :=
   by
   have h3 : ∀ (t : Finset ℕ) (K : ℕ → compacts G), μ (t.sup K) ≤ t.Sum fun i => μ (K i) :=
@@ -204,8 +274,14 @@ theorem innerContent_Sup_nat [T2Space G] (U : ℕ → Opens G) :
   refine' le_trans _ (le_iSup _ (L i))
   refine' le_trans _ (le_iSup _ (h2K' i))
   rfl
-#align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_Sup_nat
-
+#align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_iSup_nat
+
+/- warning: measure_theory.content.inner_content_Union_nat -> MeasureTheory.Content.innerContent_iUnion_nat is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {{U : Nat -> (Set.{u1} G)}} (hU : forall (i : Nat), IsOpen.{u1} G _inst_1 (U i)), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 (Set.iUnion.{u1, 1} G Nat (fun (i : Nat) => U i)) (isOpen_iUnion.{u1, 1} G Nat _inst_1 (fun (i : Nat) => U i) hU))) (tsum.{0, 0} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace Nat (fun (i : Nat) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 (U i) (hU i))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {{U : Nat -> (Set.{u1} G)}} (hU : forall (i : Nat), IsOpen.{u1} G _inst_1 (U i)), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 (Set.iUnion.{u1, 1} G Nat (fun (i : Nat) => U i)) (isOpen_iUnion.{u1, 1} G Nat _inst_1 (fun (i : Nat) => U i) hU))) (tsum.{0, 0} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal Nat (fun (i : Nat) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 (U i) (hU i))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_natₓ'. -/
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `inner_content_Sup_nat`.
   It required for the API of `induced_outer_measure`. -/
@@ -216,6 +292,12 @@ theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i
   rwa [opens.supr_def] at this
 #align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 
+/- warning: measure_theory.content.inner_content_comap -> MeasureTheory.Content.innerContent_comap is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (f : Homeomorph.{u1, u1} G G _inst_1 _inst_1), (forall {{K : TopologicalSpace.Compacts.{u1} G _inst_1}}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (coeFn.{succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) (fun (_x : Homeomorph.{u1, u1} G G _inst_1 _inst_1) => G -> G) (Homeomorph.hasCoeToFun.{u1, u1} G G _inst_1 _inst_1) f) (Homeomorph.continuous.{u1, u1} G G _inst_1 _inst_1 f) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), Eq.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (coeFn.{succ u1, succ u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) (fun (_x : FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) => (TopologicalSpace.Opens.{u1} G _inst_1) -> (TopologicalSpace.Opens.{u1} G _inst_1)) (FrameHom.hasCoeToFun.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) (TopologicalSpace.Opens.comap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.toContinuousMap.{u1, u1} G G _inst_1 _inst_1 f)) U)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) (f : Homeomorph.{u1, u1} G G _inst_1 _inst_1), (forall {{K : TopologicalSpace.Compacts.{u1} G _inst_1}}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G (fun (_x : G) => G) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (EquivLike.toEmbeddingLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (Homeomorph.instEquivLikeHomeomorph.{u1, u1} G G _inst_1 _inst_1))) f) (Homeomorph.continuous.{u1, u1} G G _inst_1 _inst_1 f) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), Eq.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (FunLike.coe.{succ u1, succ u1, succ u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (fun (_x : TopologicalSpace.Opens.{u1} G _inst_1) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.309 : TopologicalSpace.Opens.{u1} G _inst_1) => TopologicalSpace.Opens.{u1} G _inst_1) _x) (sSupHomClass.toFunLike.{u1, u1, u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (CompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (CompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (FrameHomClass.tosSupHomClass.{u1, u1, u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (FrameHom.instFrameHomClassFrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)))) (TopologicalSpace.Opens.comap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.toContinuousMap.{u1, u1} G G _inst_1 _inst_1 f)) U)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comapₓ'. -/
 theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
@@ -224,15 +306,27 @@ theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
 
+/- warning: measure_theory.content.is_mul_left_invariant_inner_content -> MeasureTheory.Content.is_mul_left_invariant_innerContent is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : Group.{u1} G] [_inst_3 : TopologicalGroup.{u1} G _inst_1 _inst_2], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (TopologicalGroup.to_continuousMul.{u1} G _inst_1 _inst_2 _inst_3) g) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (g : G) (U : TopologicalSpace.Opens.{u1} G _inst_1), Eq.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (coeFn.{succ u1, succ u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) (fun (_x : FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) => (TopologicalSpace.Opens.{u1} G _inst_1) -> (TopologicalSpace.Opens.{u1} G _inst_1)) (FrameHom.hasCoeToFun.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1) (TopologicalSpace.Opens.completeLattice.{u1} G _inst_1)) (TopologicalSpace.Opens.comap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.toContinuousMap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.mulLeft.{u1} G _inst_1 _inst_2 (TopologicalGroup.to_continuousMul.{u1} G _inst_1 _inst_2 _inst_3) g))) U)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : Group.{u1} G] [_inst_3 : TopologicalGroup.{u1} G _inst_1 _inst_2], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (TopologicalGroup.toContinuousMul.{u1} G _inst_1 _inst_2 _inst_3) g) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (g : G) (U : TopologicalSpace.Opens.{u1} G _inst_1), Eq.{1} ENNReal (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (FunLike.coe.{succ u1, succ u1, succ u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (fun (_x : TopologicalSpace.Opens.{u1} G _inst_1) => (fun (x._@.Mathlib.Order.Hom.CompleteLattice._hyg.309 : TopologicalSpace.Opens.{u1} G _inst_1) => TopologicalSpace.Opens.{u1} G _inst_1) _x) (sSupHomClass.toFunLike.{u1, u1, u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (CompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (CompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (FrameHomClass.tosSupHomClass.{u1, u1, u1} (FrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (FrameHom.instFrameHomClassFrameHom.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} G _inst_1)))) (TopologicalSpace.Opens.comap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.toContinuousMap.{u1, u1} G G _inst_1 _inst_1 (Homeomorph.mulLeft.{u1} G _inst_1 _inst_2 (TopologicalGroup.toContinuousMul.{u1} G _inst_1 _inst_2 _inst_3) g))) U)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mul_left_invariant_innerContentₓ'. -/
 @[to_additive]
-theorem is_mulLeft_invariant_innerContent [Group G] [TopologicalGroup G]
+theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
     (U : Opens G) :
     μ.innerContent (Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) = μ.innerContent U := by
   convert μ.inner_content_comap (Homeomorph.mulLeft g) (fun K => h g) U
-#align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mulLeft_invariant_innerContent
-#align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_inner_content
-
+#align measure_theory.content.is_mul_left_invariant_inner_content MeasureTheory.Content.is_mul_left_invariant_innerContent
+#align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_innerContent
+
+/- warning: measure_theory.content.inner_content_pos_of_is_mul_left_invariant -> MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.to_continuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1), (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), (Set.Nonempty.{u1} G ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) -> (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))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.toContinuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1), (Ne.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), (Set.Nonempty.{u1} G (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) -> (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))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ U))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariantₓ'. -/
 @[to_additive]
 theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -250,8 +344,14 @@ theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [Topolog
     (rel_iSup_sum μ.inner_content μ.inner_content_bot (· ≤ ·) μ.inner_content_Sup_nat _ _).trans _
   simp only [μ.is_mul_left_invariant_inner_content h3, Finset.sum_const, nsmul_eq_mul, le_refl]
 #align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
-#align measure_theory.content.inner_content_pos_of_is_add_left_invariant MeasureTheory.Content.inner_content_pos_of_is_add_left_invariant
-
+#align measure_theory.content.inner_content_pos_of_is_add_left_invariant MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant
+
+/- warning: measure_theory.content.inner_content_mono' -> MeasureTheory.Content.innerContent_mono' is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {{U : Set.{u1} G}} {{V : Set.{u1} G}} (hU : IsOpen.{u1} G _inst_1 U) (hV : IsOpen.{u1} G _inst_1 V), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) U V) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 V hV)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) {{U : Set.{u1} G}} {{V : Set.{u1} G}} (hU : IsOpen.{u1} G _inst_1 U) (hV : IsOpen.{u1} G _inst_1 V), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) U V) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)) (MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 V hV)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.inner_content_mono' MeasureTheory.Content.innerContent_mono'ₓ'. -/
 theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
   biSup_mono fun K hK => hK.trans h2
@@ -259,27 +359,45 @@ theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2
 
 section OuterMeasure
 
+#print MeasureTheory.Content.outerMeasure /-
 /-- Extending a content on compact sets to an outer measure on all sets. -/
 protected def outerMeasure : OuterMeasure G :=
   inducedOuterMeasure (fun U hU => μ.innerContent ⟨U, hU⟩) isOpen_empty μ.innerContent_bot
 #align measure_theory.content.outer_measure MeasureTheory.Content.outerMeasure
+-/
 
 variable [T2Space G]
 
+#print MeasureTheory.Content.outerMeasure_opens /-
 theorem outerMeasure_opens (U : Opens G) : μ.OuterMeasure U = μ.innerContent U :=
   inducedOuterMeasure_eq' (fun _ => isOpen_iUnion) μ.innerContent_iUnion_nat μ.innerContent_mono U.2
 #align measure_theory.content.outer_measure_opens MeasureTheory.Content.outerMeasure_opens
+-/
 
+#print MeasureTheory.Content.outerMeasure_of_isOpen /-
 theorem outerMeasure_of_isOpen (U : Set G) (hU : IsOpen U) :
     μ.OuterMeasure U = μ.innerContent ⟨U, hU⟩ :=
   μ.outerMeasure_opens ⟨U, hU⟩
 #align measure_theory.content.outer_measure_of_is_open MeasureTheory.Content.outerMeasure_of_isOpen
+-/
 
+/- warning: measure_theory.content.outer_measure_le -> MeasureTheory.Content.outerMeasure_le is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (U : TopologicalSpace.Opens.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{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.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (U : TopologicalSpace.Opens.{u1} G _inst_1) (K : TopologicalSpace.Compacts.{u1} G _inst_1), (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_le MeasureTheory.Content.outerMeasure_leₓ'. -/
 theorem outerMeasure_le (U : Opens G) (K : Compacts G) (hUK : (U : Set G) ⊆ K) :
     μ.OuterMeasure U ≤ μ K :=
   (μ.outerMeasure_opens U).le.trans <| μ.innerContent_le U K hUK
 #align measure_theory.content.outer_measure_le MeasureTheory.Content.outerMeasure_le
 
+/- warning: measure_theory.content.le_outer_measure_compacts -> MeasureTheory.Content.le_outerMeasure_compacts is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (K : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (K : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compactsₓ'. -/
 theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :=
   by
   rw [content.outer_measure, induced_outer_measure_eq_infi]
@@ -288,15 +406,33 @@ theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :
   · exact μ.inner_content_mono
 #align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compacts
 
+/- warning: measure_theory.content.outer_measure_eq_infi -> MeasureTheory.Content.outerMeasure_eq_iInf is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (A : Set.{u1} G), Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) (iInf.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} G) (fun (U : Set.{u1} G) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsOpen.{u1} G _inst_1 U) (fun (hU : IsOpen.{u1} G _inst_1 U) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) A U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) A U) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (A : Set.{u1} G), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) (iInf.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} G) (fun (U : Set.{u1} G) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsOpen.{u1} G _inst_1 U) (fun (hU : IsOpen.{u1} G _inst_1 U) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) A U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) A U) => MeasureTheory.Content.innerContent.{u1} G _inst_1 μ (TopologicalSpace.Opens.mk.{u1} G _inst_1 U hU)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInfₓ'. -/
 theorem outerMeasure_eq_iInf (A : Set G) :
     μ.OuterMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (h : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
   inducedOuterMeasure_eq_iInf _ μ.innerContent_iUnion_nat μ.innerContent_mono A
 #align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInf
 
+/- warning: measure_theory.content.outer_measure_interior_compacts -> MeasureTheory.Content.outerMeasure_interior_compacts is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (K : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toHasLe.{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.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (interior.{u1} G _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (K : TopologicalSpace.Compacts.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (interior.{u1} G _inst_1 (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_interior_compacts MeasureTheory.Content.outerMeasure_interior_compactsₓ'. -/
 theorem outerMeasure_interior_compacts (K : Compacts G) : μ.OuterMeasure (interior K) ≤ μ K :=
   (μ.outerMeasure_opens <| Opens.interior K).le.trans <| μ.innerContent_le _ _ interior_subset
 #align measure_theory.content.outer_measure_interior_compacts MeasureTheory.Content.outerMeasure_interior_compacts
 
+/- warning: measure_theory.content.outer_measure_exists_compact -> MeasureTheory.Content.outerMeasure_exists_compact is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {U : TopologicalSpace.Opens.{u1} G _inst_1}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (LE.le.{0} ENNReal (Preorder.toHasLe.{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.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.setLike.{u1} G _inst_1)))) K)) ((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))) ε))))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {U : TopologicalSpace.Opens.{u1} G _inst_1}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Compacts.{u1} G _inst_1) G (TopologicalSpace.Compacts.instSetLikeCompacts.{u1} G _inst_1) K)) (ENNReal.some ε))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compactₓ'. -/
 theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure K + ε :=
   by
@@ -305,6 +441,12 @@ theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.OuterMeasure U ≠ 
   exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outer_measure_compacts K) _⟩
 #align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compact
 
+/- warning: measure_theory.content.outer_measure_exists_open -> MeasureTheory.Content.outerMeasure_exists_open is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {A : Set.{u1} G}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Exists.{succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (fun (U : TopologicalSpace.Opens.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) A ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (LE.le.{0} ENNReal (Preorder.toHasLe.{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.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) ((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))) ε))))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] {A : Set.{u1} G}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Exists.{succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (fun (U : TopologicalSpace.Opens.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) A (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A) (ENNReal.some ε))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_openₓ'. -/
 theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {ε : ℝ≥0} (hε : ε ≠ 0) :
     ∃ U : Opens G, A ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure A + ε :=
   by
@@ -314,6 +456,12 @@ theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {
   exact ⟨⟨U, hU⟩, h2U, h3U⟩; swap; exact μ.inner_content_Union_nat
 #align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_open
 
+/- warning: measure_theory.content.outer_measure_preimage -> MeasureTheory.Content.outerMeasure_preimage is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (f : Homeomorph.{u1, u1} G G _inst_1 _inst_1), (forall {{K : TopologicalSpace.Compacts.{u1} G _inst_1}}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (coeFn.{succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) (fun (_x : Homeomorph.{u1, u1} G G _inst_1 _inst_1) => G -> G) (Homeomorph.hasCoeToFun.{u1, u1} G G _inst_1 _inst_1) f) (Homeomorph.continuous.{u1, u1} G G _inst_1 _inst_1 f) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (A : Set.{u1} G), Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Set.preimage.{u1, u1} G G (coeFn.{succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) (fun (_x : Homeomorph.{u1, u1} G G _inst_1 _inst_1) => G -> G) (Homeomorph.hasCoeToFun.{u1, u1} G G _inst_1 _inst_1) f) A)) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (f : Homeomorph.{u1, u1} G G _inst_1 _inst_1), (forall {{K : TopologicalSpace.Compacts.{u1} G _inst_1}}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G (fun (_x : G) => G) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (EquivLike.toEmbeddingLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (Homeomorph.instEquivLikeHomeomorph.{u1, u1} G G _inst_1 _inst_1))) f) (Homeomorph.continuous.{u1, u1} G G _inst_1 _inst_1 f) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (A : Set.{u1} G), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Set.preimage.{u1, u1} G G (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G (fun (_x : G) => G) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (EquivLike.toEmbeddingLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} G G _inst_1 _inst_1) G G (Homeomorph.instEquivLikeHomeomorph.{u1, u1} G G _inst_1 _inst_1))) f) A)) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimageₓ'. -/
 theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (A : Set G) : μ.OuterMeasure (f ⁻¹' A) = μ.OuterMeasure A :=
   by
@@ -323,6 +471,12 @@ theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ
   intro s hs; convert μ.inner_content_comap f h ⟨s, hs⟩
 #align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimage
 
+/- warning: measure_theory.content.outer_measure_lt_top_of_is_compact -> MeasureTheory.Content.outerMeasure_lt_top_of_isCompact is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : LocallyCompactSpace.{u1} G _inst_1] {K : Set.{u1} G}, (IsCompact.{u1} G _inst_1 K) -> (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.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) K) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : LocallyCompactSpace.{u1} G _inst_1] {K : Set.{u1} G}, (IsCompact.{u1} G _inst_1 K) -> (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} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) K) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompactₓ'. -/
 theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK : IsCompact K) :
     μ.OuterMeasure K < ∞ :=
   by
@@ -335,14 +489,26 @@ theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK
     
 #align measure_theory.content.outer_measure_lt_top_of_is_compact MeasureTheory.Content.outerMeasure_lt_top_of_isCompact
 
+/- warning: measure_theory.content.is_mul_left_invariant_outer_measure -> MeasureTheory.Content.is_mul_left_invariant_outerMeasure is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.to_continuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (g : G) (A : Set.{u1} G), Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Set.preimage.{u1, u1} G G (fun (h : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g h) A)) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.toContinuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (g : G) (A : Set.{u1} G), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Set.preimage.{u1, u1} G G (fun (h : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g h) A)) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) A))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.is_mul_left_invariant_outer_measure MeasureTheory.Content.is_mul_left_invariant_outerMeasureₓ'. -/
 @[to_additive]
 theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
     (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
     (A : Set G) : μ.OuterMeasure ((fun h => g * h) ⁻¹' A) = μ.OuterMeasure A := by
   convert μ.outer_measure_preimage (Homeomorph.mulLeft g) (fun K => h g) A
 #align measure_theory.content.is_mul_left_invariant_outer_measure MeasureTheory.Content.is_mul_left_invariant_outerMeasure
-#align measure_theory.content.is_add_left_invariant_outer_measure MeasureTheory.Content.is_add_left_invariant_outer_measure
-
+#align measure_theory.content.is_add_left_invariant_outer_measure MeasureTheory.Content.is_add_left_invariant_outerMeasure
+
+/- warning: measure_theory.content.outer_measure_caratheodory -> MeasureTheory.Content.outerMeasure_caratheodory is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (A : Set.{u1} G), Iff (MeasurableSet.{u1} G (MeasureTheory.OuterMeasure.caratheodory.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ)) A) (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Inter.inter.{u1} (Set.{u1} G) (Set.hasInter.{u1} G) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U) A)) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} G) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} G) (Set.booleanAlgebra.{u1} G)) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U) A))) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.setLike.{u1} G _inst_1)))) U)))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] (A : Set.{u1} G), Iff (MeasurableSet.{u1} G (MeasureTheory.OuterMeasure.caratheodory.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ)) A) (forall (U : TopologicalSpace.Opens.{u1} G _inst_1), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (Inter.inter.{u1} (Set.{u1} G) (Set.instInterSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U) A)) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} G) (Set.instSDiffSet.{u1} G) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U) A))) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} G _inst_1) G (TopologicalSpace.Opens.instSetLikeOpens.{u1} G _inst_1) U)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodoryₓ'. -/
 theorem outerMeasure_caratheodory (A : Set G) :
     measurable_set[μ.OuterMeasure.caratheodory] A ↔
       ∀ U : Opens G, μ.OuterMeasure (U ∩ A) + μ.OuterMeasure (U \ A) ≤ μ.OuterMeasure U :=
@@ -353,6 +519,12 @@ theorem outerMeasure_caratheodory (A : Set G) :
   apply inner_content_mono'
 #align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodory
 
+/- warning: measure_theory.content.outer_measure_pos_of_is_mul_left_invariant -> MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.to_continuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K)) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1), (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (forall {U : Set.{u1} G}, (IsOpen.{u1} G _inst_1 U) -> (Set.Nonempty.{u1} G U) -> (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))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (coeFn.{succ u1, succ u1} (MeasureTheory.OuterMeasure.{u1} G) (fun (_x : MeasureTheory.OuterMeasure.{u1} G) => (Set.{u1} G) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{u1} G) (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) U))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1) [_inst_2 : T2Space.{u1} G _inst_1] [_inst_3 : Group.{u1} G] [_inst_4 : TopologicalGroup.{u1} G _inst_1 _inst_3], (forall (g : G) {K : TopologicalSpace.Compacts.{u1} G _inst_1}, Eq.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) (TopologicalSpace.Compacts.map.{u1, u1} G G _inst_1 _inst_1 (fun (b : G) => HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3))))) g b) (continuous_mul_left.{u1} G _inst_1 (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_3)))) (TopologicalGroup.toContinuousMul.{u1} G _inst_1 _inst_3 _inst_4) g) K)) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K)) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1), (Ne.{1} ENNReal ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (forall {U : Set.{u1} G}, (IsOpen.{u1} G _inst_1 U) -> (Set.Nonempty.{u1} G U) -> (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))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) (MeasureTheory.OuterMeasure.measureOf.{u1} G (MeasureTheory.Content.outerMeasure.{u1} G _inst_1 μ) U))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.outer_measure_pos_of_is_mul_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariantₓ'. -/
 @[to_additive]
 theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
@@ -361,12 +533,13 @@ theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
   convert μ.inner_content_pos_of_is_mul_left_invariant h3 K hK ⟨U, h1U⟩ h2U
   exact μ.outer_measure_opens ⟨U, h1U⟩
 #align measure_theory.content.outer_measure_pos_of_is_mul_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant
-#align measure_theory.content.outer_measure_pos_of_is_add_left_invariant MeasureTheory.Content.outer_measure_pos_of_is_add_left_invariant
+#align measure_theory.content.outer_measure_pos_of_is_add_left_invariant MeasureTheory.Content.outerMeasure_pos_of_is_add_left_invariant
 
 variable [S : MeasurableSpace G] [BorelSpace G]
 
 include S
 
+#print MeasureTheory.Content.borel_le_caratheodory /-
 /-- For the outer measure coming from a content, all Borel sets are measurable. -/
 theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   by
@@ -399,16 +572,22 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   refine' le_trans (ge_of_eq _) (μ.le_inner_content _ _ this)
   exact μ.sup_disjoint _ _ hM.2.symm
 #align measure_theory.content.borel_le_caratheodory MeasureTheory.Content.borel_le_caratheodory
+-/
 
+#print MeasureTheory.Content.measure /-
 /-- The measure induced by the outer measure coming from a content, on the Borel sigma-algebra. -/
 protected def measure : Measure G :=
   μ.OuterMeasure.toMeasure μ.borel_le_caratheodory
 #align measure_theory.content.measure MeasureTheory.Content.measure
+-/
 
+#print MeasureTheory.Content.measure_apply /-
 theorem measure_apply {s : Set G} (hs : MeasurableSet s) : μ.Measure s = μ.OuterMeasure s :=
   toMeasure_apply _ _ hs
 #align measure_theory.content.measure_apply MeasureTheory.Content.measure_apply
+-/
 
+#print MeasureTheory.Content.regular /-
 /-- In a locally compact space, any measure constructed from a content is regular. -/
 instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
   by
@@ -432,18 +611,27 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
   refine' ⟨K, hKU, K.2, hr.trans_le _⟩
   exact (μ.le_outer_measure_compacts K).trans (le_to_measure_apply _ _ _)
 #align measure_theory.content.regular MeasureTheory.Content.regular
+-/
 
 end OuterMeasure
 
 section RegularContents
 
+#print MeasureTheory.Content.ContentRegular /-
 /-- A content `μ` is called regular if for every compact set `K`,
   `μ(K) = inf {μ(K') : K ⊂ int K' ⊂ K'`. See Paul Halmos (1950), Measure Theory, §54-/
 def ContentRegular :=
   ∀ ⦃K : TopologicalSpace.Compacts G⦄,
     μ K = ⨅ (K' : TopologicalSpace.Compacts G) (hK : (K : Set G) ⊆ interior (K' : Set G)), μ K'
 #align measure_theory.content.content_regular MeasureTheory.Content.ContentRegular
+-/
 
+/- warning: measure_theory.content.content_regular_exists_compact -> MeasureTheory.Content.contentRegular_exists_compact is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), (MeasureTheory.Content.ContentRegular.{u1} G _inst_1 μ) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1) {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K' : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K) (interior.{u1} G _inst_1 (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K'))) (LE.le.{0} ENNReal (Preorder.toHasLe.{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.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K') (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{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)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Content.{u1} G _inst_1) (fun (_x : MeasureTheory.Content.{u1} G _inst_1) => (TopologicalSpace.Compacts.{u1} G _inst_1) -> ENNReal) (MeasureTheory.Content.hasCoeToFun.{u1} G _inst_1) μ K) ((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))) ε))))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} G] (μ : MeasureTheory.Content.{u1} G _inst_1), (MeasureTheory.Content.ContentRegular.{u1} G _inst_1 μ) -> (forall (K : TopologicalSpace.Compacts.{u1} G _inst_1) {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Exists.{succ u1} (TopologicalSpace.Compacts.{u1} G _inst_1) (fun (K' : TopologicalSpace.Compacts.{u1} G _inst_1) => And (HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K) (interior.{u1} G _inst_1 (TopologicalSpace.Compacts.carrier.{u1} G _inst_1 K'))) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K') (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{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.instCanonicallyOrderedCommSemiringENNReal)))))))) ((fun (s : TopologicalSpace.Compacts.{u1} G _inst_1) => ENNReal.some (MeasureTheory.Content.toFun.{u1} G _inst_1 μ s)) K) (ENNReal.some ε))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compactₓ'. -/
 theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
     {ε : NNReal} (hε : ε ≠ 0) :
     ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε :=
@@ -463,6 +651,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
 
 variable [MeasurableSpace G] [T2Space G] [BorelSpace G]
 
+#print MeasureTheory.Content.measure_eq_content_of_regular /-
 /-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
   on compact sets.-/
 theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
@@ -485,6 +674,7 @@ theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular
   · rw [μ.measure_apply (IsCompact.measurableSet K.is_compact)]
     exact μ.le_outer_measure_compacts K
 #align measure_theory.content.measure_eq_content_of_regular MeasureTheory.Content.measure_eq_content_of_regular
+-/
 
 end RegularContents
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -128,24 +128,24 @@ def innerContent (U : Opens G) : ℝ≥0∞ :=
 
 theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
     μ K ≤ μ.innerContent U :=
-  le_supᵢ_of_le K <| le_supᵢ _ h2
+  le_iSup_of_le K <| le_iSup _ h2
 #align measure_theory.content.le_inner_content MeasureTheory.Content.le_innerContent
 
 theorem innerContent_le (U : Opens G) (K : Compacts G) (h2 : (U : Set G) ⊆ K) :
     μ.innerContent U ≤ μ K :=
-  supᵢ₂_le fun K' hK' => μ.mono _ _ (Subset.trans hK' h2)
+  iSup₂_le fun K' hK' => μ.mono _ _ (Subset.trans hK' h2)
 #align measure_theory.content.inner_content_le MeasureTheory.Content.innerContent_le
 
 theorem innerContent_of_isCompact {K : Set G} (h1K : IsCompact K) (h2K : IsOpen K) :
     μ.innerContent ⟨K, h2K⟩ = μ ⟨K, h1K⟩ :=
-  le_antisymm (supᵢ₂_le fun K' hK' => μ.mono _ ⟨K, h1K⟩ hK') (μ.le_innerContent _ _ Subset.rfl)
+  le_antisymm (iSup₂_le fun K' hK' => μ.mono _ ⟨K, h1K⟩ hK') (μ.le_innerContent _ _ Subset.rfl)
 #align measure_theory.content.inner_content_of_is_compact MeasureTheory.Content.innerContent_of_isCompact
 
 theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
   by
   refine' le_antisymm _ (zero_le _)
   rw [← μ.empty]
-  refine' supᵢ₂_le fun K hK => _
+  refine' iSup₂_le fun K hK => _
   have : K = ⊥ := by
     ext1
     rw [subset_empty_iff.mp hK, compacts.coe_bot]
@@ -156,7 +156,7 @@ theorem innerContent_bot : μ.innerContent ⊥ = 0 :=
 /-- This is "unbundled", because that it required for the API of `induced_outer_measure`. -/
 theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
-  bsupᵢ_mono fun K hK => hK.trans h2
+  biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono MeasureTheory.Content.innerContent_mono
 
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
@@ -169,7 +169,7 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
   conv at this =>
     rhs
     rw [inner_content];
-  simp only [lt_supᵢ_iff] at this
+  simp only [lt_iSup_iff] at this
   rcases this with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
   rw [← tsub_le_iff_right]; exact le_of_lt h2U
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
@@ -187,7 +187,7 @@ theorem innerContent_Sup_nat [T2Space G] (U : ℕ → Opens G) :
     · intro n s hn ih
       rw [Finset.sup_insert, Finset.sum_insert hn]
       exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
-  refine' supᵢ₂_le fun K hK => _
+  refine' iSup₂_le fun K hK => _
   obtain ⟨t, ht⟩ := K.is_compact.elim_finite_subcover _ (fun i => (U i).IsOpen) _
   swap
   · rwa [← opens.coe_supr]
@@ -197,29 +197,29 @@ theorem innerContent_Sup_nat [T2Space G] (U : ℕ → Opens G) :
   let L : ℕ → compacts G := fun n => ⟨K' n, h1K' n⟩
   convert le_trans (h3 t L) _
   · ext1
-    rw [compacts.coe_finset_sup, Finset.sup_eq_supᵢ]
+    rw [compacts.coe_finset_sup, Finset.sup_eq_iSup]
     exact h3K'
   refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)
   intro i hi
-  refine' le_trans _ (le_supᵢ _ (L i))
-  refine' le_trans _ (le_supᵢ _ (h2K' i))
+  refine' le_trans _ (le_iSup _ (L i))
+  refine' le_trans _ (le_iSup _ (h2K' i))
   rfl
 #align measure_theory.content.inner_content_Sup_nat MeasureTheory.Content.innerContent_Sup_nat
 
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `inner_content_Sup_nat`.
   It required for the API of `induced_outer_measure`. -/
-theorem innerContent_unionᵢ_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
-    μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_unionᵢ hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ :=
+theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
+    μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ :=
   by
   have := μ.inner_content_Sup_nat fun i => ⟨U i, hU i⟩
   rwa [opens.supr_def] at this
-#align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_unionᵢ_nat
+#align measure_theory.content.inner_content_Union_nat MeasureTheory.Content.innerContent_iUnion_nat
 
 theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.Continuous) = μ K)
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
-  refine' (compacts.equiv f).Surjective.supᵢ_congr _ fun K => supᵢ_congr_Prop image_subset_iff _
+  refine' (compacts.equiv f).Surjective.iSup_congr _ fun K => iSup_congr_Prop image_subset_iff _
   intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, ENNReal.coe_eq_coe, compacts.equiv]
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
@@ -247,14 +247,14 @@ theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [Topolog
     simpa only [opens.supr_def, opens.coe_comap, Subtype.coe_mk]
   refine' (μ.le_inner_content _ _ this).trans _
   refine'
-    (rel_supᵢ_sum μ.inner_content μ.inner_content_bot (· ≤ ·) μ.inner_content_Sup_nat _ _).trans _
+    (rel_iSup_sum μ.inner_content μ.inner_content_bot (· ≤ ·) μ.inner_content_Sup_nat _ _).trans _
   simp only [μ.is_mul_left_invariant_inner_content h3, Finset.sum_const, nsmul_eq_mul, le_refl]
 #align measure_theory.content.inner_content_pos_of_is_mul_left_invariant MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
 #align measure_theory.content.inner_content_pos_of_is_add_left_invariant MeasureTheory.Content.inner_content_pos_of_is_add_left_invariant
 
 theorem innerContent_mono' ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2 : U ⊆ V) :
     μ.innerContent ⟨U, hU⟩ ≤ μ.innerContent ⟨V, hV⟩ :=
-  bsupᵢ_mono fun K hK => hK.trans h2
+  biSup_mono fun K hK => hK.trans h2
 #align measure_theory.content.inner_content_mono' MeasureTheory.Content.innerContent_mono'
 
 section OuterMeasure
@@ -267,7 +267,7 @@ protected def outerMeasure : OuterMeasure G :=
 variable [T2Space G]
 
 theorem outerMeasure_opens (U : Opens G) : μ.OuterMeasure U = μ.innerContent U :=
-  inducedOuterMeasure_eq' (fun _ => isOpen_unionᵢ) μ.innerContent_unionᵢ_nat μ.innerContent_mono U.2
+  inducedOuterMeasure_eq' (fun _ => isOpen_iUnion) μ.innerContent_iUnion_nat μ.innerContent_mono U.2
 #align measure_theory.content.outer_measure_opens MeasureTheory.Content.outerMeasure_opens
 
 theorem outerMeasure_of_isOpen (U : Set G) (hU : IsOpen U) :
@@ -283,15 +283,15 @@ theorem outerMeasure_le (U : Opens G) (K : Compacts G) (hUK : (U : Set G) ⊆ K)
 theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.OuterMeasure K :=
   by
   rw [content.outer_measure, induced_outer_measure_eq_infi]
-  · exact le_infᵢ fun U => le_infᵢ fun hU => le_infᵢ <| μ.le_inner_content K ⟨U, hU⟩
+  · exact le_iInf fun U => le_iInf fun hU => le_iInf <| μ.le_inner_content K ⟨U, hU⟩
   · exact μ.inner_content_Union_nat
   · exact μ.inner_content_mono
 #align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compacts
 
-theorem outerMeasure_eq_infᵢ (A : Set G) :
+theorem outerMeasure_eq_iInf (A : Set G) :
     μ.OuterMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (h : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
-  inducedOuterMeasure_eq_infᵢ _ μ.innerContent_unionᵢ_nat μ.innerContent_mono A
-#align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_infᵢ
+  inducedOuterMeasure_eq_iInf _ μ.innerContent_iUnion_nat μ.innerContent_mono A
+#align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInf
 
 theorem outerMeasure_interior_compacts (K : Compacts G) : μ.OuterMeasure (interior K) ≤ μ K :=
   (μ.outerMeasure_opens <| Opens.interior K).le.trans <| μ.innerContent_le _ _ interior_subset
@@ -376,21 +376,21 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   rw [μ.outer_measure_caratheodory]
   intro U'
   rw [μ.outer_measure_of_is_open ((U' : Set G) ∩ U) (U'.is_open.inter hU)]
-  simp only [inner_content, supᵢ_subtype']
+  simp only [inner_content, iSup_subtype']
   rw [opens.coe_mk]
   haveI : Nonempty { L : compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [ENNReal.supᵢ_add]
-  refine' supᵢ_le _
+  rw [ENNReal.iSup_add]
+  refine' iSup_le _
   rintro ⟨L, hL⟩
   simp only [subset_inter_iff] at hL
   have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
   refine' le_trans (add_le_add_left (μ.outer_measure.mono' this) _) _
   rw [μ.outer_measure_of_is_open (↑U' \ L) (IsOpen.sdiff U'.2 L.2.IsClosed)]
-  simp only [inner_content, supᵢ_subtype']
+  simp only [inner_content, iSup_subtype']
   rw [opens.coe_mk]
   haveI : Nonempty { M : compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [ENNReal.add_supᵢ]
-  refine' supᵢ_le _
+  rw [ENNReal.add_iSup]
+  refine' iSup_le _
   rintro ⟨M, hM⟩
   simp only [subset_diff] at hM
   have : (↑(L ⊔ M) : Set G) ⊆ U' := by
@@ -416,7 +416,7 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
     by
     refine' ⟨fun A hA r (hr : _ < _) => _⟩
     rw [μ.measure_apply hA, outer_measure_eq_infi] at hr
-    simp only [infᵢ_lt_iff] at hr
+    simp only [iInf_lt_iff] at hr
     rcases hr with ⟨U, hUo, hAU, hr⟩
     rw [← μ.outer_measure_of_is_open U hUo, ← μ.measure_apply hUo.measurable_set] at hr
     exact ⟨U, hAU, hUo, hr⟩
@@ -427,7 +427,7 @@ instance regular [LocallyCompactSpace G] : μ.Measure.regular :=
     exact μ.outer_measure_lt_top_of_is_compact hK
   refine' ⟨fun U hU r hr => _⟩
   rw [measure_apply _ hU.measurable_set, μ.outer_measure_of_is_open U hU] at hr
-  simp only [inner_content, lt_supᵢ_iff] at hr
+  simp only [inner_content, lt_iSup_iff] at hr
   rcases hr with ⟨K, hKU, hr⟩
   refine' ⟨K, hKU, K.2, hr.trans_le _⟩
   exact (μ.le_outer_measure_compacts K).trans (le_to_measure_apply _ _ _)
@@ -453,7 +453,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
   have lower_bound_infi :
     μ K + ε ≤
       ⨅ (K' : TopologicalSpace.Compacts G) (h : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
-    le_infᵢ fun K' => le_infᵢ fun K'_hyp => le_of_lt (hc K' K'_hyp)
+    le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
   rw [← H] at lower_bound_infi
   exact
     (lt_self_iff_false (μ K)).mp

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -59,7 +59,7 @@ noncomputable section
 
 open Set TopologicalSpace
 
-open NNReal Ennreal MeasureTheory
+open NNReal ENNReal MeasureTheory
 
 namespace MeasureTheory
 
@@ -111,7 +111,7 @@ theorem sup_le (K₁ K₂ : Compacts G) : μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K
 #align measure_theory.content.sup_le MeasureTheory.Content.sup_le
 
 theorem lt_top (K : Compacts G) : μ K < ∞ :=
-  Ennreal.coe_lt_top
+  ENNReal.coe_lt_top
 #align measure_theory.content.lt_top MeasureTheory.Content.lt_top
 
 theorem empty : μ ⊥ = 0 := by
@@ -162,10 +162,10 @@ theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε :=
   by
-  have h'ε := Ennreal.coe_ne_zero.2 hε
+  have h'ε := ENNReal.coe_ne_zero.2 hε
   cases le_or_lt (μ.inner_content U) ε
   · exact ⟨⊥, empty_subset _, le_add_left h⟩
-  have := Ennreal.sub_lt_self hU h.ne_bot h'ε
+  have := ENNReal.sub_lt_self hU h.ne_bot h'ε
   conv at this =>
     rhs
     rw [inner_content];
@@ -199,7 +199,7 @@ theorem innerContent_Sup_nat [T2Space G] (U : ℕ → Opens G) :
   · ext1
     rw [compacts.coe_finset_sup, Finset.sup_eq_supᵢ]
     exact h3K'
-  refine' le_trans (Finset.sum_le_sum _) (Ennreal.sum_le_tsum t)
+  refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)
   intro i hi
   refine' le_trans _ (le_supᵢ _ (L i))
   refine' le_trans _ (le_supᵢ _ (h2K' i))
@@ -220,7 +220,7 @@ theorem innerContent_comap (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K
     (U : Opens G) : μ.innerContent (Opens.comap f.toContinuousMap U) = μ.innerContent U :=
   by
   refine' (compacts.equiv f).Surjective.supᵢ_congr _ fun K => supᵢ_congr_Prop image_subset_iff _
-  intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, Ennreal.coe_eq_coe, compacts.equiv]
+  intro hK; simp only [Equiv.coe_fn_mk, Subtype.mk_eq_mk, ENNReal.coe_eq_coe, compacts.equiv]
   apply h
 #align measure_theory.content.inner_content_comap MeasureTheory.Content.innerContent_comap
 
@@ -309,7 +309,7 @@ theorem outerMeasure_exists_open {A : Set G} (hA : μ.OuterMeasure A ≠ ∞) {
     ∃ U : Opens G, A ⊆ U ∧ μ.OuterMeasure U ≤ μ.OuterMeasure A + ε :=
   by
   rcases induced_outer_measure_exists_set _ _ μ.inner_content_mono hA
-      (Ennreal.coe_ne_zero.2 hε) with
+      (ENNReal.coe_ne_zero.2 hε) with
     ⟨U, hU, h2U, h3U⟩
   exact ⟨⟨U, hU⟩, h2U, h3U⟩; swap; exact μ.inner_content_Union_nat
 #align measure_theory.content.outer_measure_exists_open MeasureTheory.Content.outerMeasure_exists_open
@@ -379,7 +379,7 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   simp only [inner_content, supᵢ_subtype']
   rw [opens.coe_mk]
   haveI : Nonempty { L : compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [Ennreal.supᵢ_add]
+  rw [ENNReal.supᵢ_add]
   refine' supᵢ_le _
   rintro ⟨L, hL⟩
   simp only [subset_inter_iff] at hL
@@ -389,7 +389,7 @@ theorem borel_le_caratheodory : S ≤ μ.OuterMeasure.caratheodory :=
   simp only [inner_content, supᵢ_subtype']
   rw [opens.coe_mk]
   haveI : Nonempty { M : compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
-  rw [Ennreal.add_supᵢ]
+  rw [ENNReal.add_supᵢ]
   refine' supᵢ_le _
   rintro ⟨M, hM⟩
   simp only [subset_diff] at hM
@@ -458,7 +458,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
   exact
     (lt_self_iff_false (μ K)).mp
       (lt_of_le_of_lt' lower_bound_infi
-        (Ennreal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ennreal.coe_ne_zero.mpr hε)))
+        (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ennreal.coe_ne_zero.mpr hε)))
 #align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compact
 
 variable [MeasurableSpace G] [T2Space G] [BorelSpace G]
@@ -469,7 +469,7 @@ theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular
     (K : TopologicalSpace.Compacts G) : μ.Measure ↑K = μ K :=
   by
   refine' le_antisymm _ _
-  · apply Ennreal.le_of_forall_pos_le_add
+  · apply ENNReal.le_of_forall_pos_le_add
     intro ε εpos content_K_finite
     obtain ⟨K', K'_hyp⟩ := content_regular_exists_compact μ H K (ne_bot_of_gt εpos)
     calc

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR accompanying #12339.

Zulip discussion

@@ -333,8 +333,8 @@ theorem outerMeasure_caratheodory (A : Set G) :
       ∀ U : Opens G, μ.outerMeasure (U ∩ A) + μ.outerMeasure (U \ A) ≤ μ.outerMeasure U := by
   rw [Opens.forall]
   apply inducedOuterMeasure_caratheodory
-  apply innerContent_iUnion_nat
-  apply innerContent_mono'
+  · apply innerContent_iUnion_nat
+  · apply innerContent_mono'
 #align measure_theory.content.outer_measure_caratheodory MeasureTheory.Content.outerMeasure_caratheodory
 
 @[to_additive]

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -228,8 +228,7 @@ theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
 theorem innerContent_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
     (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U := by
-  have : (interior (U : Set G)).Nonempty
-  rwa [U.isOpen.interior_eq]
+  have : (interior (U : Set G)).Nonempty := by rwa [U.isOpen.interior_eq]
   rcases compact_covered_by_mul_left_translates K.2 this with ⟨s, hs⟩
   suffices μ K ≤ s.card * μ.innerContent U by
     exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2

Main API changes

• Define R1Space, a.k.a. preregular space.
• Drop T2OrLocallyCompactRegularSpace.
• Generalize all existing theorems about T2OrLocallyCompactRegularSpace to R1Space.
• Drop the [T2OrLocallyCompactRegularSpace _] assumption if the space is known to be regular for other reason (e.g., because it's a topological group).

New theorems

• Specializes.not_disjoint: if x ⤳ y, then 𝓝 x and 𝓝 y aren't disjoint;
• specializes_iff_not_disjoint, Specializes.inseparable, disjoint_nhds_nhds_iff_not_inseparable, r1Space_iff_inseparable_or_disjoint_nhds: basic API about R1Spaces;
• Inducing.r1Space, R1Space.induced, R1Space.sInf, R1Space.iInf, R1Space.inf, instances for Subtype _, X × Y, and ∀ i, X i: basic instances for R1Space;
• IsCompact.mem_closure_iff_exists_inseparable, IsCompact.closure_eq_biUnion_inseparable: characterizations of the closure of a compact set in a preregular space;
• Inseparable.mem_measurableSet_iff: topologically inseparable points can't be separated by a Borel measurable set;
• IsCompact.closure_subset_measurableSet, IsCompact.measure_closure: in a preregular space, a measurable superset of a compact set includes its closure as well; as a corollary, closure K has the same measure as K.
• exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds: an auxiliary lemma extracted from a LocallyCompactPair instance;
• IsCompact.isCompact_isClosed_basis_nhds: if x admits a compact neighborhood, then it admits a basis of compact closed neighborhoods; in particular, a weakly locally compact preregular space is a locally compact regular space;
• isCompact_isClosed_basis_nhds: a version of the previous theorem for weakly locally compact spaces;
• exists_mem_nhds_isCompact_isClosed: in a locally compact regular space, each point admits a compact closed neighborhood.

Deprecated theorems

Some theorems about topological groups are true for any (pre)regular space, so we deprecate the special cases.

• exists_isCompact_isClosed_subset_isCompact_nhds_one: use new IsCompact.isCompact_isClosed_basis_nhds instead;
• instLocallyCompactSpaceOfWeaklyOfGroup, instLocallyCompactSpaceOfWeaklyOfAddGroup: are now implied by WeaklyLocallyCompactSpace.locallyCompactSpace;
• local_isCompact_isClosed_nhds_of_group, local_isCompact_isClosed_nhds_of_addGroup: use isCompact_isClosed_basis_nhds instead;
• exists_isCompact_isClosed_nhds_one, exists_isCompact_isClosed_nhds_zero: use exists_mem_nhds_isCompact_isClosed instead.

Renamed/moved theorems

For each renamed theorem, the old theorem is redefined as a deprecated alias.

• isOpen_setOf_disjoint_nhds_nhds: moved to Constructions;
• isCompact_closure_of_subset_compact -> IsCompact.closure_of_subset;
• IsCompact.measure_eq_infi_isOpen -> IsCompact.measure_eq_iInf_isOpen;
• exists_compact_superset_iff -> exists_isCompact_superset_iff;
• separatedNhds_of_isCompact_isCompact_isClosed -> SeparatedNhds.of_isCompact_isCompact_isClosed;
• separatedNhds_of_isCompact_isCompact -> SeparatedNhds.of_isCompact_isCompact;
• separatedNhds_of_finset_finset -> SeparatedNhds.of_finset_finset;
• point_disjoint_finset_opens_of_t2 -> SeparatedNhds.of_singleton_finset;
• separatedNhds_of_isCompact_isClosed -> SeparatedNhds.of_isCompact_isClosed;
• exists_open_superset_and_isCompact_closure -> exists_isOpen_superset_and_isCompact_closure;
• exists_open_with_compact_closure -> exists_isOpen_mem_isCompact_closure;

@@ -41,9 +41,9 @@ For `μ : Content G`, we define
 * `μ.outerMeasure` : the outer measure associated to `μ`.
 * `μ.measure`      : the Borel measure associated to `μ`.
 
-These definitions are given for spaces which are either T2, or locally compact and regular (which
-covers possibly non-Hausdorff locally compact groups). The resulting measure `μ.measure` is always
-outer regular by design. When the space is locally compact, `μ.measure` is also regular.
+These definitions are given for spaces which are R₁.
+The resulting measure `μ.measure` is always outer regular by design.
+When the space is locally compact, `μ.measure` is also regular.
 
 ## References
 
@@ -171,7 +171,7 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
 
 /-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
-theorem innerContent_iSup_nat [T2OrLocallyCompactRegularSpace G] (U : ℕ → Opens G) :
+theorem innerContent_iSup_nat [R1Space G] (U : ℕ → Opens G) :
     μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
   have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
     intro t K
@@ -200,7 +200,7 @@ theorem innerContent_iSup_nat [T2OrLocallyCompactRegularSpace G] (U : ℕ → Op
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `innerContent_iSup_nat`.
   It is required for the API of `inducedOuterMeasure`. -/
-theorem innerContent_iUnion_nat [T2OrLocallyCompactRegularSpace G] ⦃U : ℕ → Set G⦄
+theorem innerContent_iUnion_nat [R1Space G] ⦃U : ℕ → Set G⦄
     (hU : ∀ i : ℕ, IsOpen (U i)) :
     μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
   have := μ.innerContent_iSup_nat fun i => ⟨U i, hU i⟩
@@ -225,8 +225,7 @@ theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
 #align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_innerContent
 
 @[to_additive]
-theorem innerContent_pos_of_is_mul_left_invariant [T2OrLocallyCompactRegularSpace G] [Group G]
-    [TopologicalGroup G]
+theorem innerContent_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
     (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U := by
   have : (interior (U : Set G)).Nonempty
@@ -255,7 +254,7 @@ protected def outerMeasure : OuterMeasure G :=
   inducedOuterMeasure (fun U hU => μ.innerContent ⟨U, hU⟩) isOpen_empty μ.innerContent_bot
 #align measure_theory.content.outer_measure MeasureTheory.Content.outerMeasure
 
-variable [T2OrLocallyCompactRegularSpace G]
+variable [R1Space G]
 
 theorem outerMeasure_opens (U : Opens G) : μ.outerMeasure U = μ.innerContent U :=
   inducedOuterMeasure_eq' (fun _ => isOpen_iUnion) μ.innerContent_iUnion_nat μ.innerContent_mono U.2
@@ -450,7 +449,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
     (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ENNReal.coe_ne_zero.mpr hε)))
 #align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compact
 
-variable [MeasurableSpace G] [T2OrLocallyCompactRegularSpace G] [BorelSpace G]
+variable [MeasurableSpace G] [R1Space G] [BorelSpace G]
 
 /-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
   on compact sets. -/

The construction we have is given in T2 spaces, but it works in non-Hausdorff spaces modulo a few modifications.

For this, we introduce an ad hoc class T2OrLocallyCompactRegularSpace, which is just enough to unify the arguments, as a replacement for the class ClosableCompactSubsetOpenSpace (which is not strong enough). In the file Separation.lean, we move some material that was only available on T2 spaces to this new class.

The construction is needed for a forthcoming improvement of uniqueness results for Haar measures, based on https://mathoverflow.net/questions/456670/uniqueness-of-left-invariant-borel-probability-measure-on-compact-groups.

@@ -41,7 +41,9 @@ For `μ : Content G`, we define
 * `μ.outerMeasure` : the outer measure associated to `μ`.
 * `μ.measure`      : the Borel measure associated to `μ`.
 
-We prove that, on a locally compact space, the measure `μ.measure` is regular.
+These definitions are given for spaces which are either T2, or locally compact and regular (which
+covers possibly non-Hausdorff locally compact groups). The resulting measure `μ.measure` is always
+outer regular by design. When the space is locally compact, `μ.measure` is also regular.
 
 ## References
 
@@ -68,7 +70,8 @@ structure Content (G : Type w) [TopologicalSpace G] where
   toFun : Compacts G → ℝ≥0
   mono' : ∀ K₁ K₂ : Compacts G, (K₁ : Set G) ⊆ K₂ → toFun K₁ ≤ toFun K₂
   sup_disjoint' :
-    ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → toFun (K₁ ⊔ K₂) = toFun K₁ + toFun K₂
+    ∀ K₁ K₂ : Compacts G, Disjoint (K₁ : Set G) K₂ → IsClosed (K₁ : Set G) → IsClosed (K₂ : Set G)
+      → toFun (K₁ ⊔ K₂) = toFun K₁ + toFun K₂
   sup_le' : ∀ K₁ K₂ : Compacts G, toFun (K₁ ⊔ K₂) ≤ toFun K₁ + toFun K₂
 #align measure_theory.content MeasureTheory.Content
 
@@ -96,7 +99,8 @@ theorem mono (K₁ K₂ : Compacts G) (h : (K₁ : Set G) ⊆ K₂) : μ K₁ 
   simp [apply_eq_coe_toFun, μ.mono' _ _ h]
 #align measure_theory.content.mono MeasureTheory.Content.mono
 
-theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂) :
+theorem sup_disjoint (K₁ K₂ : Compacts G) (h : Disjoint (K₁ : Set G) K₂)
+    (h₁ : IsClosed (K₁ : Set G)) (h₂ : IsClosed (K₂ : Set G)) :
     μ (K₁ ⊔ K₂) = μ K₁ + μ K₂ := by
   simp [apply_eq_coe_toFun, μ.sup_disjoint' _ _ h]
 #align measure_theory.content.sup_disjoint MeasureTheory.Content.sup_disjoint
@@ -167,7 +171,7 @@ theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ 
 #align measure_theory.content.inner_content_exists_compact MeasureTheory.Content.innerContent_exists_compact
 
 /-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
-theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
+theorem innerContent_iSup_nat [T2OrLocallyCompactRegularSpace G] (U : ℕ → Opens G) :
     μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
   have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
     intro t K
@@ -196,7 +200,8 @@ theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
 /-- The inner content of a union of sets is at most the sum of the individual inner contents.
   This is the "unbundled" version of `innerContent_iSup_nat`.
   It is required for the API of `inducedOuterMeasure`. -/
-theorem innerContent_iUnion_nat [T2Space G] ⦃U : ℕ → Set G⦄ (hU : ∀ i : ℕ, IsOpen (U i)) :
+theorem innerContent_iUnion_nat [T2OrLocallyCompactRegularSpace G] ⦃U : ℕ → Set G⦄
+    (hU : ∀ i : ℕ, IsOpen (U i)) :
     μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by
   have := μ.innerContent_iSup_nat fun i => ⟨U i, hU i⟩
   rwa [Opens.iSup_def] at this
@@ -220,7 +225,8 @@ theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
 #align measure_theory.content.is_add_left_invariant_inner_content MeasureTheory.Content.is_add_left_invariant_innerContent
 
 @[to_additive]
-theorem innerContent_pos_of_is_mul_left_invariant [T2Space G] [Group G] [TopologicalGroup G]
+theorem innerContent_pos_of_is_mul_left_invariant [T2OrLocallyCompactRegularSpace G] [Group G]
+    [TopologicalGroup G]
     (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
     (hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U := by
   have : (interior (U : Set G)).Nonempty
@@ -249,7 +255,7 @@ protected def outerMeasure : OuterMeasure G :=
   inducedOuterMeasure (fun U hU => μ.innerContent ⟨U, hU⟩) isOpen_empty μ.innerContent_bot
 #align measure_theory.content.outer_measure MeasureTheory.Content.outerMeasure
 
-variable [T2Space G]
+variable [T2OrLocallyCompactRegularSpace G]
 
 theorem outerMeasure_opens (U : Opens G) : μ.outerMeasure U = μ.innerContent U :=
   inducedOuterMeasure_eq' (fun _ => isOpen_iUnion) μ.innerContent_iUnion_nat μ.innerContent_mono U.2
@@ -358,22 +364,34 @@ theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
   rw [ENNReal.iSup_add]
   refine' iSup_le _
   rintro ⟨L, hL⟩
+  let L' : Compacts G := ⟨closure L, L.isCompact.closure⟩
+  suffices μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure U' by
+    have A : μ L ≤ μ L' := μ.mono _ _ subset_closure
+    exact (add_le_add_right A _).trans this
   simp only [subset_inter_iff] at hL
-  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
+  have hL'U : (L' : Set G) ⊆ U := IsCompact.closure_subset_of_isOpen L.2 hU hL.2
+  have hL'U' : (L' : Set G) ⊆ (U' : Set G) := IsCompact.closure_subset_of_isOpen L.2 U'.2 hL.1
+  have : ↑U' \ U ⊆ U' \ L' := diff_subset_diff_right hL'U
   refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
-  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
+  rw [μ.outerMeasure_of_isOpen (↑U' \ L') (IsOpen.sdiff U'.2 isClosed_closure)]
   simp only [innerContent, iSup_subtype']
   rw [Opens.coe_mk]
-  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
+  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ closure L } := ⟨⟨⊥, empty_subset _⟩⟩
   rw [ENNReal.add_iSup]
   refine' iSup_le _
   rintro ⟨M, hM⟩
-  simp only [subset_diff] at hM
-  have : (↑(L ⊔ M) : Set G) ⊆ U' := by
-    simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
+  let M' : Compacts G := ⟨closure M, M.isCompact.closure⟩
+  suffices μ L' + μ M' ≤ μ.outerMeasure U' by
+    have A : μ M ≤ μ M' := μ.mono _ _ subset_closure
+    exact (add_le_add_left A _).trans this
+  have hM' : (M' : Set G) ⊆ U' \ L' :=
+    IsCompact.closure_subset_of_isOpen M.2 (IsOpen.sdiff U'.2 isClosed_closure) hM
+  have : (↑(L' ⊔ M') : Set G) ⊆ U' := by
+    simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and]
+    exact hM'.trans (diff_subset _ _ )
   rw [μ.outerMeasure_of_isOpen (↑U') U'.2]
   refine' le_trans (ge_of_eq _) (μ.le_innerContent _ _ this)
-  exact μ.sup_disjoint _ _ hM.2.symm
+  exact μ.sup_disjoint L' M' (subset_diff.1 hM').2.symm isClosed_closure isClosed_closure
 #align measure_theory.content.borel_le_caratheodory MeasureTheory.Content.borel_le_caratheodory
 
 /-- The measure induced by the outer measure coming from a content, on the Borel sigma-algebra. -/
@@ -385,19 +403,21 @@ theorem measure_apply {s : Set G} (hs : MeasurableSet s) : μ.measure s = μ.out
   toMeasure_apply _ _ hs
 #align measure_theory.content.measure_apply MeasureTheory.Content.measure_apply
 
+instance outerRegular : μ.measure.OuterRegular := by
+  refine ⟨fun A hA r (hr : _ < _) ↦ ?_⟩
+  rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
+  simp only [iInf_lt_iff] at hr
+  rcases hr with ⟨U, hUo, hAU, hr⟩
+  rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
+  exact ⟨U, hAU, hUo, hr⟩
+
 /-- In a locally compact space, any measure constructed from a content is regular. -/
 instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
-  have : μ.measure.OuterRegular := by
-    refine' ⟨fun A hA r (hr : _ < _) => _⟩
-    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
-    simp only [iInf_lt_iff] at hr
-    rcases hr with ⟨U, hUo, hAU, hr⟩
-    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
-    exact ⟨U, hAU, hUo, hr⟩
   have : IsFiniteMeasureOnCompacts μ.measure := by
     refine' ⟨fun K hK => _⟩
-    rw [measure_apply _ hK.measurableSet]
-    exact μ.outerMeasure_lt_top_of_isCompact hK
+    apply (measure_mono subset_closure).trans_lt _
+    rw [measure_apply _ isClosed_closure.measurableSet]
+    exact μ.outerMeasure_lt_top_of_isCompact hK.closure
   refine' ⟨fun U hU r hr => _⟩
   rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
   simp only [innerContent, lt_iSup_iff] at hr
@@ -430,7 +450,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
     (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ENNReal.coe_ne_zero.mpr hε)))
 #align measure_theory.content.content_regular_exists_compact MeasureTheory.Content.contentRegular_exists_compact
 
-variable [MeasurableSpace G] [T2Space G] [BorelSpace G]
+variable [MeasurableSpace G] [T2OrLocallyCompactRegularSpace G] [BorelSpace G]
 
 /-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
   on compact sets. -/
@@ -441,16 +461,17 @@ theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular
     intro ε εpos _
     obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
     calc
-      μ.measure ↑K ≤ μ.measure (interior ↑K') := by
-        rw [μ.measure_apply isOpen_interior.measurableSet,
-          μ.measure_apply K.isCompact.measurableSet]
-        exact μ.outerMeasure.mono K'_hyp.left
+      μ.measure ↑K ≤ μ.measure (interior ↑K') := measure_mono K'_hyp.1
       _ ≤ μ K' := by
         rw [μ.measure_apply (IsOpen.measurableSet isOpen_interior)]
         exact μ.outerMeasure_interior_compacts K'
       _ ≤ μ K + ε := K'_hyp.right
-  · rw [μ.measure_apply (IsCompact.measurableSet K.isCompact)]
-    exact μ.le_outerMeasure_compacts K
+  · calc
+    μ K ≤ μ ⟨closure K, K.2.closure⟩ := μ.mono _ _ subset_closure
+    _ ≤ μ.measure (closure K) := by
+      rw [μ.measure_apply (isClosed_closure.measurableSet)]
+      exact μ.le_outerMeasure_compacts _
+    _ = μ.measure K := Measure.OuterRegular.measure_closure_eq_of_isCompact K.2
 #align measure_theory.content.measure_eq_content_of_regular MeasureTheory.Content.measure_eq_content_of_regular
 
 end RegularContents

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.

@@ -157,7 +157,7 @@ theorem innerContent_mono ⦃U V : Set G⦄ (hU : IsOpen U) (hV : IsOpen V) (h2
 theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
   have h'ε := ENNReal.coe_ne_zero.2 hε
-  cases' le_or_lt (μ.innerContent U) ε with h h
+  rcases le_or_lt (μ.innerContent U) ε with h | h
   · exact ⟨⊥, empty_subset _, le_add_left h⟩
   have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
   conv at h₂ => rhs; rw [innerContent]

@@ -305,7 +305,8 @@ theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ
   convert μ.innerContent_comap f h ⟨s, hs⟩
 #align measure_theory.content.outer_measure_preimage MeasureTheory.Content.outerMeasure_preimage
 
-theorem outerMeasure_lt_top_of_isCompact [LocallyCompactSpace G] {K : Set G} (hK : IsCompact K) :
+theorem outerMeasure_lt_top_of_isCompact [WeaklyLocallyCompactSpace G]
+    {K : Set G} (hK : IsCompact K) :
     μ.outerMeasure K < ∞ := by
   rcases exists_compact_superset hK with ⟨F, h1F, h2F⟩
   calc
@@ -385,7 +386,7 @@ theorem measure_apply {s : Set G} (hs : MeasurableSet s) : μ.measure s = μ.out
 #align measure_theory.content.measure_apply MeasureTheory.Content.measure_apply
 
 /-- In a locally compact space, any measure constructed from a content is regular. -/
-instance regular [LocallyCompactSpace G] : μ.measure.Regular := by
+instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
   have : μ.measure.OuterRegular := by
     refine' ⟨fun A hA r (hr : _ < _) => _⟩
     rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
-
-! This file was ported from Lean 3 source module measure_theory.measure.content
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Measure.MeasureSpace
 import Mathlib.MeasureTheory.Measure.Regular
 import Mathlib.Topology.Sets.Compacts
 
+#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
+
 /-!
 # Contents
 

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -287,7 +287,7 @@ theorem outerMeasure_interior_compacts (K : Compacts G) : μ.outerMeasure (inter
 
 theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.outerMeasure U ≠ ∞) {ε : ℝ≥0}
     (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.outerMeasure U ≤ μ.outerMeasure K + ε := by
-  rw [μ.outerMeasure_opens] at hU⊢
+  rw [μ.outerMeasure_opens] at hU ⊢
   rcases μ.innerContent_exists_compact hU hε with ⟨K, h1K, h2K⟩
   exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outerMeasure_compacts K) _⟩
 #align measure_theory.content.outer_measure_exists_compact MeasureTheory.Content.outerMeasure_exists_compact

@@ -123,7 +123,7 @@ theorem empty : μ ⊥ = 0 := by
   obtain a function defined on all open sets, by taking the supremum of the content of all compact
   subsets. -/
 def innerContent (U : Opens G) : ℝ≥0∞ :=
-  ⨆ (K : Compacts G) (_h : (K : Set G) ⊆ U), μ K
+  ⨆ (K : Compacts G) (_ : (K : Set G) ⊆ U), μ K
 #align measure_theory.content.inner_content MeasureTheory.Content.innerContent
 
 theorem le_innerContent (K : Compacts G) (U : Opens G) (h2 : (K : Set G) ⊆ U) :
@@ -277,7 +277,7 @@ theorem le_outerMeasure_compacts (K : Compacts G) : μ K ≤ μ.outerMeasure K :
 #align measure_theory.content.le_outer_measure_compacts MeasureTheory.Content.le_outerMeasure_compacts
 
 theorem outerMeasure_eq_iInf (A : Set G) :
-    μ.outerMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (_h : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
+    μ.outerMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (_ : A ⊆ U), μ.innerContent ⟨U, hU⟩ :=
   inducedOuterMeasure_eq_iInf _ μ.innerContent_iUnion_nat μ.innerContent_mono A
 #align measure_theory.content.outer_measure_eq_infi MeasureTheory.Content.outerMeasure_eq_iInf
 
@@ -416,7 +416,7 @@ section RegularContents
   `μ(K) = inf {μ(K') : K ⊂ int K' ⊂ K'}`. See Paul Halmos (1950), Measure Theory, §54-/
 def ContentRegular :=
   ∀ ⦃K : TopologicalSpace.Compacts G⦄,
-    μ K = ⨅ (K' : TopologicalSpace.Compacts G) (_hK : (K : Set G) ⊆ interior (K' : Set G)), μ K'
+    μ K = ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K'
 #align measure_theory.content.content_regular MeasureTheory.Content.ContentRegular
 
 theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
@@ -425,7 +425,7 @@ theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSp
   by_contra hc
   simp only [not_exists, not_and, not_le] at hc
   have lower_bound_iInf : μ K + ε ≤
-      ⨅ (K' : TopologicalSpace.Compacts G) (_h : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
+      ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
     le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
   rw [← H] at lower_bound_iInf
   exact (lt_self_iff_false (μ K)).mp (lt_of_le_of_lt' lower_bound_iInf

I have misported is_foo to Foo because I misunderstood the rule for IsLawfulFoo. This PR recover Is of Foo which is ported from is_foo. This PR also renames some misported theorems.

@@ -396,7 +396,7 @@ instance regular [LocallyCompactSpace G] : μ.measure.Regular := by
     rcases hr with ⟨U, hUo, hAU, hr⟩
     rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
     exact ⟨U, hAU, hUo, hr⟩
-  have : FiniteMeasureOnCompacts μ.measure := by
+  have : IsFiniteMeasureOnCompacts μ.measure := by
     refine' ⟨fun K hK => _⟩
     rw [measure_apply _ hK.measurableSet]
     exact μ.outerMeasure_lt_top_of_isCompact hK