measure_theory.measure.regular ⟷ Mathlib.MeasureTheory.Measure.Regular

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

@@ -140,7 +140,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegularWRT /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
@@ -158,7 +158,7 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup /-
 theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
@@ -169,7 +169,7 @@ theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add /-
 theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
@@ -219,7 +219,7 @@ end InnerRegular
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
 #print MeasureTheory.Measure.OuterRegular /-
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
 This definition implies the same equality for any (not necessarily measurable) set, see
@@ -272,7 +272,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_of_lt /-
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
@@ -298,7 +298,7 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_add /-
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
@@ -306,7 +306,7 @@ theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ 
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_le_add /-
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
@@ -318,7 +318,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print MeasurableSet.exists_isOpen_diff_lt /-
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -356,7 +356,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 
 end OuterRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
 #print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /-
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
@@ -403,7 +403,7 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 #print MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
@@ -430,9 +430,9 @@ theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOp
 
 open Finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 #print MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
@@ -545,7 +545,7 @@ instance zero : Regular (0 : Measure α) :=
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isCompact /-
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -577,7 +577,7 @@ theorem innerRegularWRT_measurable [Regular μ] :
   Regular.innerRegularWRT.measurableSet_of_isOpen isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegularWRT_measurable
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_lt_add /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -587,7 +587,7 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_diff_lt /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
@@ -604,7 +604,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isCompact_of_ne_top /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
@@ -614,7 +614,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isCompact_of_ne_top /-
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
@@ -662,7 +662,7 @@ end Regular
 
 namespace WeaklyRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -671,7 +671,7 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.measure_eq_iSup_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
@@ -688,7 +688,7 @@ theorem innerRegular_measurable [WeaklyRegular μ] :
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 #print MeasurableSet.exists_isClosed_lt_add /-
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
@@ -698,7 +698,7 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 #print MeasurableSet.exists_isClosed_diff_lt /-
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -712,7 +712,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -723,7 +723,7 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/

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

@@ -189,7 +189,7 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
     (hB₁ : ∀ K, pb K → MeasurableSet K) (hB₂ : ∀ U, qb U → MeasurableSet U) :
     InnerRegularWRT (map f μ) pb qb := by
   intro U hU r hr
-  rw [map_apply_of_ae_measurable hf (hB₂ _ hU)] at hr 
+  rw [map_apply_of_ae_measurable hf (hB₂ _ hU)] at hr
   rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
   refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
@@ -200,7 +200,7 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
 theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q :=
   by
   intro U hU r hr
-  rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr 
+  rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegularWRT.smul
 -/
@@ -335,7 +335,7 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular :=
   by
   refine' ⟨fun A hA r hr => _⟩
-  rw [map_apply f.measurable hA, ← f.image_symm] at hr 
+  rw [map_apply f.measurable hA, ← f.image_symm] at hr
   rcases Set.exists_isOpen_lt_of_lt _ r hr with ⟨U, hAU, hUo, hU⟩
   have : IsOpen (f.symm ⁻¹' U) := hUo.preimage f.symm.continuous
   refine' ⟨f.symm ⁻¹' U, image_subset_iff.1 hAU, this, _⟩
@@ -349,7 +349,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
   rcases eq_or_ne x 0 with (rfl | h0)
   · rw [zero_smul]; exact outer_regular.zero
   · refine' ⟨fun A hA r hr => _⟩
-    rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr 
+    rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr
     simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
 -/
@@ -378,7 +378,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
         (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
     rw [← inter_Union, iUnion_disjointed, s.spanning, inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
-  rw [lt_tsub_iff_right, add_comm] at hδε 
+  rw [lt_tsub_iff_right, add_comm] at hδε
   have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n :=
     by
     intro n
@@ -386,7 +386,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by rw [H₁];
       exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).Ne
     rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
-    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU 
+    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
     exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
   choose U hAU hUo hU
   refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_iUnion hUo, _⟩
@@ -453,7 +453,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     rcases exists_between hr with ⟨r', hsr', hr'r⟩
     rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with ⟨-, -, U, hsU, -, hUo, -, H⟩
     refine' ⟨U, hsU, hUo, _⟩
-    rw [add_tsub_cancel_of_le hsr'.le] at H ; exact H.trans_lt hr'r
+    rw [add_tsub_cancel_of_le hsr'.le] at H; exact H.trans_lt hr'r
   refine' MeasurableSet.induction_on_open _ _ _
   /- The proof is by measurable induction: we should check that the property is true for the empty
     set, for open sets, and is stable by taking the complement and by taking countable disjoint
@@ -511,7 +511,7 @@ theorem of_pseudoMetrizableSpace {X : Type _} [PseudoEMetricSpace X] [Measurable
   by
   intro U hU r hr
   rcases hU.exists_Union_is_closed with ⟨F, F_closed, -, rfl, F_mono⟩
-  rw [measure_Union_eq_supr F_mono.directed_le] at hr 
+  rw [measure_Union_eq_supr F_mono.directed_le] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
 #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace
@@ -530,7 +530,7 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
     by
     rw [← measure_Union_eq_supr, hBU]
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
-  rw [this] at hr ; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
+  rw [this] at hr; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
 #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed
 -/

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

@@ -404,10 +404,10 @@ namespace InnerRegular
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-#print MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open /-
+#print MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
-theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen) (h0 : p ∅)
+theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen) (h0 : p ∅)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
     InnerRegularWRT μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   by
@@ -425,7 +425,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen
     _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
     _ ≤ μ (K \ U') + ε + ε := by mono*; exacts [hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
-#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open
+#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen
 -/
 
 open Finset
@@ -574,7 +574,7 @@ compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem innerRegularWRT_measurable [Regular μ] :
     InnerRegularWRT μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  Regular.innerRegularWRT.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
+  Regular.innerRegularWRT.measurableSet_of_isOpen isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegularWRT_measurable
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
@@ -683,7 +683,7 @@ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : M
 #print MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable /-
 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
+  WeaklyRegular.innerRegular.measurableSet_of_isOpen isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 -/

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

@@ -141,16 +141,16 @@ namespace MeasureTheory
 namespace Measure
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
-#print MeasureTheory.Measure.InnerRegular /-
+#print MeasureTheory.Measure.InnerRegularWRT /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
 of measure greater than `r`.
 
 This definition is used to prove some facts about regular and weakly regular measures without
 repeating the proofs. -/
-def InnerRegular {α} {m : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
+def InnerRegularWRT {α} {m : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
   ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ (K : _) (_ : K ⊆ U), p K ∧ r < μ K
-#align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegular
+#align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegularWRT
 -/
 
 namespace InnerRegular
@@ -159,19 +159,19 @@ variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α
   {ε : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
-#print MeasureTheory.Measure.InnerRegular.measure_eq_iSup /-
-theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
+#print MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup /-
+theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
   by
   refine'
     le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
-#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
+#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
-#print MeasureTheory.Measure.InnerRegular.exists_subset_lt_add /-
-theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
+#print MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add /-
+theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
   by
   cases' eq_or_ne (μ U) 0 with h₀ h₀
@@ -179,39 +179,39 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
     rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
   · rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
     exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
-#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
+#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add
 -/
 
-#print MeasureTheory.Measure.InnerRegular.map /-
+#print MeasureTheory.Measure.InnerRegularWRT.map /-
 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
-    (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
+    (H : InnerRegularWRT μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
     (hB₁ : ∀ K, pb K → MeasurableSet K) (hB₂ : ∀ U, qb U → MeasurableSet U) :
-    InnerRegular (map f μ) pb qb := by
+    InnerRegularWRT (map f μ) pb qb := by
   intro U hU r hr
   rw [map_apply_of_ae_measurable hf (hB₂ _ hU)] at hr 
   rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
   refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
-#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map
+#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegularWRT.map
 -/
 
-#print MeasureTheory.Measure.InnerRegular.smul /-
-theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q :=
+#print MeasureTheory.Measure.InnerRegularWRT.smul /-
+theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q :=
   by
   intro U hU r hr
   rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr 
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
-#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
+#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegularWRT.smul
 -/
 
-#print MeasureTheory.Measure.InnerRegular.trans /-
-theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
-    InnerRegular μ p q' := by
+#print MeasureTheory.Measure.InnerRegularWRT.trans /-
+theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
+    InnerRegularWRT μ p q' := by
   intro U hU r hr
   rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
   exact ⟨K, hKF.trans hFU, hpK, hrK⟩
-#align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegular.trans
+#align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegularWRT.trans
 -/
 
 end InnerRegular
@@ -239,7 +239,7 @@ class OuterRegular (μ : Measure α) : Prop where
     `μ(U) = sup {μ(K) | K ⊆ U compact}` for `U` open. -/
 @[protect_proj]
 class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
-  InnerRegular : InnerRegular μ IsCompact IsOpen
+  InnerRegularWRT : InnerRegularWRT μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
 -/
 
@@ -250,7 +250,7 @@ class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegul
     `μ(U) = sup {μ(F) | F ⊆ U compact}` for `U` open. -/
 @[protect_proj]
 class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
-  InnerRegular : InnerRegular μ IsClosed IsOpen
+  InnerRegularWRT : InnerRegularWRT μ IsClosed IsOpen
 #align measure_theory.measure.weakly_regular MeasureTheory.Measure.WeaklyRegular
 -/
 
@@ -258,8 +258,8 @@ class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
 -- see Note [lower instance priority]
 /-- A regular measure is weakly regular. -/
 instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : WeaklyRegular μ
-    where InnerRegular U hU r hr :=
-    let ⟨K, hKU, hcK, hK⟩ := Regular.innerRegular hU r hr
+    where InnerRegularWRT U hU r hr :=
+    let ⟨K, hKU, hcK, hK⟩ := Regular.innerRegularWRT hU r hr
     ⟨K, hKU, hcK.IsClosed, hK⟩
 #align measure_theory.measure.regular.weakly_regular MeasureTheory.Measure.Regular.weaklyRegular
 -/
@@ -404,12 +404,12 @@ namespace InnerRegular
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-#print MeasureTheory.Measure.InnerRegular.measurableSet_of_open /-
+#print MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
-theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
+theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen) (h0 : p ∅)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
-    InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
+    InnerRegularWRT μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   by
   rintro s ⟨hs, hμs⟩ r hr
   obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
@@ -425,7 +425,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
     _ ≤ μ (K \ U') + ε + ε := by mono*; exacts [hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
-#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
+#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open
 -/
 
 open Finset
@@ -433,11 +433,11 @@ open Finset
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » s) -/
-#print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
+#print MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
 theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
-    (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ :=
+    (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ :=
   by
   have hfin : ∀ {s}, μ s ≠ ⊤ := measure_ne_top μ
   suffices
@@ -449,7 +449,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     by
     refine'
       { OuterRegular := fun s hs r hr => _
-        InnerRegular := H }
+        InnerRegularWRT := H }
     rcases exists_between hr with ⟨r', hsr', hr'r⟩
     rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with ⟨-, -, U, hsU, -, hUo, -, H⟩
     refine' ⟨U, hsU, hUo, _⟩
@@ -500,27 +500,27 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
         _ ≤ ∑' n, (μ (s n) + δ n) := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
-#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite
+#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite
 -/
 
-#print MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace /-
+#print MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace /-
 /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
 by closed sets. -/
-theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
-    (μ : Measure X) : InnerRegular μ IsClosed IsOpen :=
+theorem of_pseudoMetrizableSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
+    (μ : Measure X) : InnerRegularWRT μ IsClosed IsOpen :=
   by
   intro U hU r hr
   rcases hU.exists_Union_is_closed with ⟨F, F_closed, -, rfl, F_mono⟩
   rw [measure_Union_eq_supr F_mono.directed_le] at hr 
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
-#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
+#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace
 -/
 
-#print MeasureTheory.Measure.InnerRegular.isCompact_isClosed /-
+#print MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed /-
 /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/
 theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X]
-    [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed :=
+    [MeasurableSpace X] (μ : Measure X) : InnerRegularWRT μ IsCompact IsClosed :=
   by
   intro F hF r hr
   set B : ℕ → Set X := compactCovering X
@@ -532,7 +532,7 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
   rw [this] at hr ; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
-#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
+#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed
 -/
 
 end InnerRegular
@@ -550,7 +550,7 @@ instance zero : Regular (0 : Measure α) :=
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (K : _) (_ : K ⊆ U), IsCompact K ∧ r < μ K :=
-  Regular.innerRegular hU r hr
+  Regular.innerRegularWRT hU r hr
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
 -/
 
@@ -558,7 +558,7 @@ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U)
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
 theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
     μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K :=
-  Regular.innerRegular.measure_eq_iSup hU
+  Regular.innerRegularWRT.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 -/
 
@@ -569,15 +569,13 @@ theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 -/
 
-#print MeasureTheory.Measure.Regular.innerRegular_measurable /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
-theorem innerRegular_measurable [Regular μ] :
-    InnerRegular μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
-#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
--/
+theorem innerRegularWRT_measurable [Regular μ] :
+    InnerRegularWRT μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
+  Regular.innerRegularWRT.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
+#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegularWRT_measurable
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_lt_add /-
@@ -585,7 +583,7 @@ theorem innerRegular_measurable [Regular μ] :
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ μ A < μ K + ε :=
-  Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
+  Regular.innerRegularWRT_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 -/
 
@@ -612,7 +610,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ r < μ K :=
-  Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
+  Regular.innerRegularWRT_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 -/
 
@@ -622,7 +620,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 inside by compact sets. -/
 theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsCompact K), μ K :=
-  Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
+  Regular.innerRegularWRT_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 -/
 
@@ -684,7 +682,7 @@ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : M
 
 #print MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable /-
 theorem innerRegular_measurable [WeaklyRegular μ] :
-    InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
+    InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
@@ -735,10 +733,10 @@ theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
 -/
 
-#print MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet /-
+#print MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top /-
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
-theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
+theorem restrict_of_measure_ne_top [BorelSpace α] [WeaklyRegular μ] (A : Set α)
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) :=
   by
   haveI : Fact (μ A < ∞) := ⟨h'A.lt_top⟩
@@ -749,33 +747,33 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
     ⟨F, hFVA, hFc, hF⟩
   refine' ⟨F, hFVA.trans (inter_subset_left _ _), hFc, _⟩
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
-#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
+#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top
 -/
 
-#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure /-
+#print MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_of_isFiniteMeasure /-
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
+instance (priority := 100) of_pseudoMetrizableSpace_of_isFiniteMeasure {X : Type _}
     [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
     WeaklyRegular μ :=
-  (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
+  (InnerRegularWRT.of_pseudoMetrizableSpace μ).weaklyRegular_of_finite μ
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_of_isFiniteMeasure
 -/
 
-#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite /-
+#print MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_secondCountable_of_locallyFinite /-
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
-instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _}
-    [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
-    [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
+instance (priority := 100) of_pseudoMetrizableSpace_secondCountable_of_locallyFinite {X : Type _}
+    [PseudoEMetricSpace X] [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
     by
     refine' (μ.finite_spanning_sets_in_open'.mono' fun U hU => _).OuterRegular
     have : Fact (μ U < ∞) := ⟨hU.2⟩
     exact ⟨hU.1, inferInstance⟩
   ⟨inner_regular.of_pseudo_emetric_space μ⟩
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_secondCountable_of_locallyFinite
 -/
 
 end WeaklyRegular
@@ -790,7 +788,8 @@ instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasur
     [IsLocallyFiniteMeasure μ] : Regular μ
     where
   lt_top_of_isCompact K hK := hK.measure_lt_top
-  InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
+  InnerRegularWRT :=
+    (InnerRegularWRT.isCompact_isClosed μ).trans (InnerRegularWRT.of_pseudoMetrizableSpace μ)
 #align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

@@ -565,7 +565,7 @@ theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ :
 #print MeasureTheory.Measure.Regular.exists_compact_not_null /-
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
-    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+    ENNReal.iSup_eq_zero, Classical.not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/19c869efa56bbb8b500f2724c0b77261edbfa28c

@@ -648,15 +648,16 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • 
 #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
 -/
 
-#print MeasureTheory.Measure.Regular.sigmaFinite /-
+#print MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts /-
 -- see Note [lower instance priority]
 /-- A regular measure in a σ-compact space is σ-finite. -/
-instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : SigmaFinite μ :=
+instance (priority := 100) MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts
+    [SigmaCompactSpace α] [Regular μ] : SigmaFinite μ :=
   ⟨⟨{   Set := compactCovering α
         set_mem := fun n => trivial
         Finite := fun n => (isCompact_compactCovering α n).measure_lt_top
         spanning := iUnion_compactCovering α }⟩⟩
-#align measure_theory.measure.regular.sigma_finite MeasureTheory.Measure.Regular.sigmaFinite
+#align measure_theory.measure.regular.sigma_finite MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts
 -/
 
 end Regular

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

@@ -3,7 +3,7 @@ Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
 -/
-import Mathbin.MeasureTheory.Constructions.BorelSpace.Basic
+import MeasureTheory.Constructions.BorelSpace.Basic
 
 #align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
 
@@ -140,7 +140,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
@@ -158,7 +158,7 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular.measure_eq_iSup /-
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
@@ -169,7 +169,7 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular.exists_subset_lt_add /-
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
@@ -219,7 +219,7 @@ end InnerRegular
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
 #print MeasureTheory.Measure.OuterRegular /-
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
 This definition implies the same equality for any (not necessarily measurable) set, see
@@ -272,7 +272,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_of_lt /-
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
@@ -298,7 +298,7 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_add /-
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
@@ -306,7 +306,7 @@ theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ 
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_le_add /-
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
@@ -318,7 +318,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print MeasurableSet.exists_isOpen_diff_lt /-
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -356,7 +356,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 
 end OuterRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
 #print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /-
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
@@ -403,7 +403,7 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 #print MeasureTheory.Measure.InnerRegular.measurableSet_of_open /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
@@ -430,9 +430,9 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
 
 open Finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 #print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
@@ -545,7 +545,7 @@ instance zero : Regular (0 : Measure α) :=
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isCompact /-
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -579,7 +579,7 @@ theorem innerRegular_measurable [Regular μ] :
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_lt_add /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -589,7 +589,7 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_diff_lt /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
@@ -606,7 +606,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isCompact_of_ne_top /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
@@ -616,7 +616,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isCompact_of_ne_top /-
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
@@ -663,7 +663,7 @@ end Regular
 
 namespace WeaklyRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -672,7 +672,7 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.measure_eq_iSup_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
@@ -689,7 +689,7 @@ theorem innerRegular_measurable [WeaklyRegular μ] :
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 #print MeasurableSet.exists_isClosed_lt_add /-
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
@@ -699,7 +699,7 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 #print MeasurableSet.exists_isClosed_diff_lt /-
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -713,7 +713,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -724,7 +724,7 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -484,7 +484,8 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
     refine'
       ⟨⋃ k ∈ t, F k, Union_mono fun k => Union_subset fun _ => hFs _, ⋃ n, U n, Union_mono hsU,
-        isClosed_biUnion t.finite_to_set fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
+        Set.Finite.isClosed_biUnion t.finite_to_set fun k _ => hFc k, isOpen_iUnion hUo,
+        ht.le.trans _, _⟩
     · calc
         ∑ k in t, μ (s k) + ε / 2 ≤ ∑ k in t, μ (F k) + ∑ k in t, δ k + ε / 2 := by
           rw [← sum_add_distrib]; exact add_le_add_right (sum_le_sum fun k hk => hF k) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -413,7 +413,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
   by
   rintro s ⟨hs, hμs⟩ r hr
   obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
-    use (μ s - r) / 2; simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
+    use(μ s - r) / 2; simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_is_open_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
   rcases(U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
   replace hsU' := diff_subset_comm.1 hsU'

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

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
-
-! This file was ported from Lean 3 source module measure_theory.measure.regular
-! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Constructions.BorelSpace.Basic
 
+#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
+
 /-!
 # Regular measures
 
@@ -143,7 +140,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
@@ -161,7 +158,7 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular.measure_eq_iSup /-
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
@@ -172,7 +169,7 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular.exists_subset_lt_add /-
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
@@ -222,7 +219,7 @@ end InnerRegular
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
 #print MeasureTheory.Measure.OuterRegular /-
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
 This definition implies the same equality for any (not necessarily measurable) set, see
@@ -275,7 +272,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_of_lt /-
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
@@ -301,7 +298,7 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_lt_add /-
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
@@ -309,7 +306,7 @@ theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ 
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print Set.exists_isOpen_le_add /-
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
@@ -321,7 +318,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 #print MeasurableSet.exists_isOpen_diff_lt /-
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -359,7 +356,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 
 end OuterRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
 #print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /-
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
@@ -406,7 +403,7 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 #print MeasureTheory.Measure.InnerRegular.measurableSet_of_open /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
@@ -433,9 +430,9 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
 
 open Finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 #print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
@@ -547,7 +544,7 @@ instance zero : Regular (0 : Measure α) :=
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isCompact /-
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -581,7 +578,7 @@ theorem innerRegular_measurable [Regular μ] :
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_lt_add /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -591,7 +588,7 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_isCompact_diff_lt /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
@@ -608,7 +605,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isCompact_of_ne_top /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
@@ -618,7 +615,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isCompact_of_ne_top /-
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
@@ -665,7 +662,7 @@ end Regular
 
 namespace WeaklyRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.exists_lt_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -674,7 +671,7 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 #print IsOpen.measure_eq_iSup_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
@@ -691,7 +688,7 @@ theorem innerRegular_measurable [WeaklyRegular μ] :
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 #print MeasurableSet.exists_isClosed_lt_add /-
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
@@ -701,7 +698,7 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 #print MeasurableSet.exists_isClosed_diff_lt /-
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -715,7 +712,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.exists_lt_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -726,7 +723,7 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 #print MeasurableSet.measure_eq_iSup_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/

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

@@ -162,6 +162,7 @@ variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α
   {ε : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+#print MeasureTheory.Measure.InnerRegular.measure_eq_iSup /-
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
   by
@@ -169,8 +170,10 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+#print MeasureTheory.Measure.InnerRegular.exists_subset_lt_add /-
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
   by
@@ -180,7 +183,9 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
   · rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
     exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
 #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
+-/
 
+#print MeasureTheory.Measure.InnerRegular.map /-
 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
     (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
@@ -192,13 +197,16 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
   refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
 #align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map
+-/
 
+#print MeasureTheory.Measure.InnerRegular.smul /-
 theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q :=
   by
   intro U hU r hr
   rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr 
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
+-/
 
 #print MeasureTheory.Measure.InnerRegular.trans /-
 theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
@@ -268,6 +276,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+#print Set.exists_isOpen_lt_of_lt /-
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
 theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r) :
@@ -278,7 +287,9 @@ theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0
     ⟨U, hAU, hUo, hU⟩
   exact ⟨U, (subset_to_measurable _ _).trans hAU, hUo, hU⟩
 #align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_lt
+-/
 
+#print Set.measure_eq_iInf_isOpen /-
 /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets
 containing it. -/
 theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
@@ -288,14 +299,18 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
   refine' le_of_forall_lt' fun r hr => _
   simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+#print Set.exists_isOpen_lt_add /-
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+#print Set.exists_isOpen_le_add /-
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
   by
@@ -304,8 +319,10 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
   · rcases A.exists_is_open_lt_add H hε with ⟨U, AU, U_open, hU⟩
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+#print MeasurableSet.exists_isOpen_diff_lt /-
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε :=
@@ -314,7 +331,9 @@ theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA :
   use U, hAU, hUo, hU.trans_le le_top
   exact measure_diff_lt_of_lt_add hA hAU hA' hU
 #align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_lt
+-/
 
+#print MeasureTheory.Measure.OuterRegular.map /-
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β]
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular :=
   by
@@ -325,7 +344,9 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
   refine' ⟨f.symm ⁻¹' U, image_subset_iff.1 hAU, this, _⟩
   rwa [map_apply f.measurable this.measurable_set, f.preimage_symm, f.preimage_image]
 #align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.map
+-/
 
+#print MeasureTheory.Measure.OuterRegular.smul /-
 protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) :
     (x • μ).OuterRegular := by
   rcases eq_or_ne x 0 with (rfl | h0)
@@ -334,6 +355,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
     rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr 
     simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
+-/
 
 end OuterRegular
 
@@ -385,6 +407,7 @@ namespace InnerRegular
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+#print MeasureTheory.Measure.InnerRegular.measurableSet_of_open /-
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
 theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
@@ -406,6 +429,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     _ ≤ μ (K \ U') + ε + ε := by mono*; exacts [hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
 #align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
+-/
 
 open Finset
 
@@ -524,23 +548,30 @@ instance zero : Regular (0 : Measure α) :=
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+#print IsOpen.exists_lt_isCompact /-
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (K : _) (_ : K ⊆ U), IsCompact K ∧ r < μ K :=
   Regular.innerRegular hU r hr
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
+-/
 
+#print IsOpen.measure_eq_iSup_isCompact /-
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
 theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
     μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K :=
   Regular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
+-/
 
+#print MeasureTheory.Measure.Regular.exists_compact_not_null /-
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
+-/
 
+#print MeasureTheory.Measure.Regular.innerRegular_measurable /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -548,16 +579,20 @@ theorem innerRegular_measurable [Regular μ] :
     InnerRegular μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.exists_isCompact_lt_add /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ μ A < μ K + ε :=
   Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.exists_isCompact_diff_lt /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -571,23 +606,29 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
       measure_diff_lt_of_lt_add hKc.measurable_set hKA (ne_top_of_le_ne_top h'A <| measure_mono hKA)
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.exists_lt_isCompact_of_ne_top /-
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ r < μ K :=
   Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.measure_eq_iSup_isCompact_of_ne_top /-
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
 theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsCompact K), μ K :=
   Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
+-/
 
+#print MeasureTheory.Measure.Regular.map /-
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
     [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).regular :=
   by
@@ -598,13 +639,16 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
         (fun U hU => hU.Preimage f.continuous) (fun K hK => hK.image f.continuous)
         (fun K hK => hK.MeasurableSet) fun U hU => hU.MeasurableSet⟩
 #align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map
+-/
 
+#print MeasureTheory.Measure.Regular.smul /-
 protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).regular :=
   by
   haveI := outer_regular.smul μ hx
   haveI := is_finite_measure_on_compacts.smul μ hx
   exact ⟨regular.inner_regular.smul x⟩
 #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
+-/
 
 #print MeasureTheory.Measure.Regular.sigmaFinite /-
 -- see Note [lower instance priority]
@@ -622,34 +666,43 @@ end Regular
 namespace WeaklyRegular
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+#print IsOpen.exists_lt_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (F : _) (_ : F ⊆ U), IsClosed F ∧ r < μ F :=
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+#print IsOpen.measure_eq_iSup_isClosed /-
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
     [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (h : IsClosed F), μ F :=
   WeaklyRegular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
+-/
 
+#print MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable /-
 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+#print MeasurableSet.exists_isClosed_lt_add /-
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
 theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs : MeasurableSet s)
     (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ s), IsClosed K ∧ μ s < μ K + ε :=
   innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+#print MeasurableSet.exists_isClosed_diff_lt /-
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (F : _) (_ : F ⊆ A), IsClosed F ∧ μ (A \ F) < ε :=
@@ -660,8 +713,10 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
       measure_diff_lt_of_lt_add hFc.measurable_set hFA (ne_top_of_le_ne_top h'A <| measure_mono hFA)
         hF⟩
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.exists_lt_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
@@ -669,15 +724,19 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
     ∃ (K : _) (_ : K ⊆ A), IsClosed K ∧ r < μ K :=
   innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+#print MeasurableSet.measure_eq_iSup_isClosed_of_ne_top /-
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsClosed K), μ K :=
   innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
+-/
 
+#print MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet /-
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
 theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
@@ -692,6 +751,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
   refine' ⟨F, hFVA.trans (inter_subset_left _ _), hFc, _⟩
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
 #align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
+-/
 
 #print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure /-
 -- see Note [lower instance priority]

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -373,8 +373,8 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
   refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_iUnion hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-    _ ≤ ∑' n, μ (A n) + δ n := (ENNReal.tsum_le_tsum fun n => (hU n).le)
-    _ = (∑' n, μ (A n)) + ∑' n, δ n := ENNReal.tsum_add
+    _ ≤ ∑' n, (μ (A n) + δ n) := (ENNReal.tsum_le_tsum fun n => (hU n).le)
+    _ = ∑' n, μ (A n) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_Union hAd hAm).symm rfl)
     _ < r := hδε
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
@@ -456,7 +456,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     -- the approximating closed set is constructed by considering finitely many sets `s i`, which
     -- cover all the measure up to `ε/2`, approximating each of these by a closed set `F i`, and
     -- taking the union of these (finitely many) `F i`.
-    have : tendsto (fun t => (∑ k in t, μ (s k)) + ε / 2) at_top (𝓝 <| μ (⋃ n, s n) + ε / 2) := by
+    have : tendsto (fun t => ∑ k in t, μ (s k) + ε / 2) at_top (𝓝 <| μ (⋃ n, s n) + ε / 2) := by
       rw [measure_Union hsd hsm]; exact tendsto.add ennreal.summable.has_sum tendsto_const_nhds
     rcases(this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
     -- the approximating open set is constructed by taking for each `s n` an approximating open set
@@ -465,9 +465,9 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
       ⟨⋃ k ∈ t, F k, Union_mono fun k => Union_subset fun _ => hFs _, ⋃ n, U n, Union_mono hsU,
         isClosed_biUnion t.finite_to_set fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
     · calc
-        (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 := by
+        ∑ k in t, μ (s k) + ε / 2 ≤ ∑ k in t, μ (F k) + ∑ k in t, δ k + ε / 2 := by
           rw [← sum_add_distrib]; exact add_le_add_right (sum_le_sum fun k hk => hF k) _
-        _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
+        _ ≤ ∑ k in t, μ (F k) + ε / 2 + ε / 2 :=
           (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _)
         _ = μ (⋃ k ∈ t, F k) + ε := _
       rw [measure_bUnion_finset, add_assoc, ENNReal.add_halves]
@@ -475,7 +475,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     ·
       calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-        _ ≤ ∑' n, μ (s n) + δ n := (ENNReal.tsum_le_tsum hU)
+        _ ≤ ∑' n, (μ (s n) + δ n) := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite

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

@@ -377,7 +377,6 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     _ = (∑' n, μ (A n)) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_Union hAd hAm).symm rfl)
     _ < r := hδε
-    
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
 -/
 
@@ -406,7 +405,6 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
     _ ≤ μ (K \ U') + ε + ε := by mono*; exacts [hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
-    
 #align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
 
 open Finset
@@ -472,7 +470,6 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
         _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
           (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _)
         _ = μ (⋃ k ∈ t, F k) + ε := _
-        
       rw [measure_bUnion_finset, add_assoc, ENNReal.add_halves]
       exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k hk => (hFc k).MeasurableSet]
     ·
@@ -481,7 +478,6 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
         _ ≤ ∑' n, μ (s n) + δ n := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
-        
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -143,7 +143,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 #print MeasureTheory.Measure.InnerRegular /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
@@ -161,7 +161,7 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
   by
@@ -170,7 +170,7 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
   by
@@ -214,7 +214,7 @@ end InnerRegular
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
 #print MeasureTheory.Measure.OuterRegular /-
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
 This definition implies the same equality for any (not necessarily measurable) set, see
@@ -267,7 +267,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
 theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r) :
@@ -289,13 +289,13 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
   simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
   by
@@ -305,7 +305,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε :=
@@ -337,7 +337,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 
 end OuterRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
 #print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /-
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
@@ -385,7 +385,7 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
 theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
@@ -411,9 +411,9 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
 
 open Finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 #print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
@@ -527,7 +527,7 @@ instance zero : Regular (0 : Measure α) :=
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (K : _) (_ : K ⊆ U), IsCompact K ∧ r < μ K :=
@@ -553,7 +553,7 @@ theorem innerRegular_measurable [Regular μ] :
   Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
@@ -561,7 +561,7 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
   Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -576,7 +576,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
@@ -584,7 +584,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
   Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
 theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
@@ -625,14 +625,14 @@ end Regular
 
 namespace WeaklyRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (F : _) (_ : F ⊆ U), IsClosed F ∧ r < μ F :=
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
     [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (h : IsClosed F), μ F :=
@@ -645,7 +645,7 @@ theorem innerRegular_measurable [WeaklyRegular μ] :
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
 theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs : MeasurableSet s)
@@ -653,7 +653,7 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
   innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (F : _) (_ : F ⊆ A), IsClosed F ∧ μ (A \ F) < ε :=
@@ -665,7 +665,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
         hF⟩
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
@@ -674,7 +674,7 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
   innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -233,7 +233,7 @@ class OuterRegular (μ : Measure α) : Prop where
   - it is inner regular for open sets, using compact sets:
     `μ(U) = sup {μ(K) | K ⊆ U compact}` for `U` open. -/
 @[protect_proj]
-class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
+class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
   InnerRegular : InnerRegular μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
 -/
@@ -342,7 +342,7 @@ end OuterRegular
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
 protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
-    (s : μ.FiniteSpanningSetsIn { U | IsOpen U ∧ OuterRegular (μ.restrict U) }) : OuterRegular μ :=
+    (s : μ.FiniteSpanningSetsIn {U | IsOpen U ∧ OuterRegular (μ.restrict U)}) : OuterRegular μ :=
   by
   refine' ⟨fun A hA r hr => _⟩
   have hm : ∀ n, MeasurableSet (s.set n) := fun n => (s.set_mem n).1.MeasurableSet
@@ -417,7 +417,7 @@ open Finset
 #print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
-theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure μ]
+theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
     (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ :=
   by
   have hfin : ∀ {s}, μ s ≠ ⊤ := measure_ne_top μ
@@ -697,14 +697,14 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
 #align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
 
-#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure /-
+#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure /-
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
-    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [FiniteMeasure μ] :
+instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
+    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
     WeaklyRegular μ :=
   (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
 -/
 
 #print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite /-
@@ -713,7 +713,7 @@ instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
 is weakly regular. -/
 instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
-    [BorelSpace X] (μ : Measure X) [LocallyFiniteMeasure μ] : WeaklyRegular μ :=
+    [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
     by
     refine' (μ.finite_spanning_sets_in_open'.mono' fun U hU => _).OuterRegular
@@ -727,16 +727,16 @@ end WeaklyRegular
 
 attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
-#print MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure /-
+#print MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure /-
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
-instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure {X : Type _}
+instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type _}
     [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
-    [LocallyFiniteMeasure μ] : Regular μ
+    [IsLocallyFiniteMeasure μ] : Regular μ
     where
   lt_top_of_isCompact K hK := hK.measure_lt_top
   InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
-#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure
+#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
 -/
 
 end Measure

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

@@ -152,7 +152,7 @@ of measure greater than `r`.
 This definition is used to prove some facts about regular and weakly regular measures without
 repeating the proofs. -/
 def InnerRegular {α} {m : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
-  ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ (K : _)(_ : K ⊆ U), p K ∧ r < μ K
+  ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ (K : _) (_ : K ⊆ U), p K ∧ r < μ K
 #align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegular
 -/
 
@@ -172,7 +172,7 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
-    (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
+    (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
   by
   cases' eq_or_ne (μ U) 0 with h₀ h₀
   · refine' ⟨∅, empty_subset _, h0, _⟩
@@ -187,7 +187,7 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
     (hB₁ : ∀ K, pb K → MeasurableSet K) (hB₂ : ∀ U, qb U → MeasurableSet U) :
     InnerRegular (map f μ) pb qb := by
   intro U hU r hr
-  rw [map_apply_of_ae_measurable hf (hB₂ _ hU)] at hr
+  rw [map_apply_of_ae_measurable hf (hB₂ _ hU)] at hr 
   rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
   refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
@@ -196,7 +196,7 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
 theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q :=
   by
   intro U hU r hr
-  rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr
+  rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr 
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
 
@@ -222,7 +222,7 @@ This definition implies the same equality for any (not necessarily measurable) s
 @[protect_proj]
 class OuterRegular (μ : Measure α) : Prop where
   OuterRegular :
-    ∀ ⦃A : Set α⦄, MeasurableSet A → ∀ r > μ A, ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < r
+    ∀ ⦃A : Set α⦄, MeasurableSet A → ∀ r > μ A, ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < r
 #align measure_theory.measure.outer_regular MeasureTheory.Measure.OuterRegular
 -/
 
@@ -271,7 +271,7 @@ instance zero : OuterRegular (0 : Measure α) :=
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
 theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r) :
-    ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < r :=
+    ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < r :=
   by
   rcases outer_regular.outer_regular (measurable_set_to_measurable μ A) r
       (by rwa [measure_to_measurable]) with
@@ -291,13 +291,13 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
-    (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
+    (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
-    (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
+    (hε : ε ≠ 0) : ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
   by
   rcases eq_or_ne (μ A) ∞ with (H | H)
   · exact ⟨univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]⟩
@@ -308,7 +308,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε :=
+    ∃ (U : _) (_ : U ⊇ A), IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε :=
   by
   rcases A.exists_is_open_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
   use U, hAU, hUo, hU.trans_le le_top
@@ -319,7 +319,7 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular :=
   by
   refine' ⟨fun A hA r hr => _⟩
-  rw [map_apply f.measurable hA, ← f.image_symm] at hr
+  rw [map_apply f.measurable hA, ← f.image_symm] at hr 
   rcases Set.exists_isOpen_lt_of_lt _ r hr with ⟨U, hAU, hUo, hU⟩
   have : IsOpen (f.symm ⁻¹' U) := hUo.preimage f.symm.continuous
   refine' ⟨f.symm ⁻¹' U, image_subset_iff.1 hAU, this, _⟩
@@ -331,7 +331,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
   rcases eq_or_ne x 0 with (rfl | h0)
   · rw [zero_smul]; exact outer_regular.zero
   · refine' ⟨fun A hA r hr => _⟩
-    rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr
+    rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr 
     simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
 
@@ -359,15 +359,15 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
         (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
     rw [← inter_Union, iUnion_disjointed, s.spanning, inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
-  rw [lt_tsub_iff_right, add_comm] at hδε
-  have : ∀ n, ∃ (U : _)(_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n :=
+  rw [lt_tsub_iff_right, add_comm] at hδε 
+  have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n :=
     by
     intro n
     have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
     have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by rw [H₁];
       exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).Ne
     rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
-    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
+    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU 
     exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
   choose U hAU hUo hU
   refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_iUnion hUo, _⟩
@@ -393,7 +393,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   by
   rintro s ⟨hs, hμs⟩ r hr
-  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
+  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
     use (μ s - r) / 2; simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_is_open_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
   rcases(U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
@@ -404,7 +404,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     μ s ≤ μ U := μ.mono hsU
     _ < μ K + ε := hKr
     _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
-    _ ≤ μ (K \ U') + ε + ε := by mono*; exacts[hμU'.le, le_rfl]
+    _ ≤ μ (K \ U') + ε + ε := by mono*; exacts [hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
     
 #align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
@@ -425,7 +425,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
     ∀ s,
       MeasurableSet s →
         ∀ (ε) (_ : ε ≠ 0),
-          ∃ (F : _)(_ : F ⊆ s)(U : _)(_ : U ⊇ s),
+          ∃ (F : _) (_ : F ⊆ s) (U : _) (_ : U ⊇ s),
             IsClosed F ∧ IsOpen U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε
     by
     refine'
@@ -434,7 +434,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
     rcases exists_between hr with ⟨r', hsr', hr'r⟩
     rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with ⟨-, -, U, hsU, -, hUo, -, H⟩
     refine' ⟨U, hsU, hUo, _⟩
-    rw [add_tsub_cancel_of_le hsr'.le] at H; exact H.trans_lt hr'r
+    rw [add_tsub_cancel_of_le hsr'.le] at H ; exact H.trans_lt hr'r
   refine' MeasurableSet.induction_on_open _ _ _
   /- The proof is by measurable induction: we should check that the property is true for the empty
     set, for open sets, and is stable by taking the complement and by taking countable disjoint
@@ -474,7 +474,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
         _ = μ (⋃ k ∈ t, F k) + ε := _
         
       rw [measure_bUnion_finset, add_assoc, ENNReal.add_halves]
-      exacts[fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k hk => (hFc k).MeasurableSet]
+      exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k hk => (hFc k).MeasurableSet]
     ·
       calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
@@ -493,7 +493,7 @@ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpa
   by
   intro U hU r hr
   rcases hU.exists_Union_is_closed with ⟨F, F_closed, -, rfl, F_mono⟩
-  rw [measure_Union_eq_supr F_mono.directed_le] at hr
+  rw [measure_Union_eq_supr F_mono.directed_le] at hr 
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
 #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
@@ -512,7 +512,7 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
     by
     rw [← measure_Union_eq_supr, hBU]
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
-  rw [this] at hr; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
+  rw [this] at hr ; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
 #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
 -/
@@ -530,7 +530,7 @@ instance zero : Regular (0 : Measure α) :=
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
-    (hr : r < μ U) : ∃ (K : _)(_ : K ⊆ U), IsCompact K ∧ r < μ K :=
+    (hr : r < μ U) : ∃ (K : _) (_ : K ⊆ U), IsCompact K ∧ r < μ K :=
   Regular.innerRegular hU r hr
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
 
@@ -557,7 +557,7 @@ theorem innerRegular_measurable [Regular μ] :
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
-    (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ A), IsCompact K ∧ μ A < μ K + ε :=
+    (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ μ A < μ K + ε :=
   Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
@@ -567,7 +567,7 @@ compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Space α] [Regular μ]
     ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ (K : _)(_ : K ⊆ A), IsCompact K ∧ μ (A \ K) < ε :=
+    ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ μ (A \ K) < ε :=
   by
   rcases hA.exists_is_compact_lt_add h'A hε with ⟨K, hKA, hKc, hK⟩
   exact
@@ -580,7 +580,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
-    (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) : ∃ (K : _)(_ : K ⊆ A), IsCompact K ∧ r < μ K :=
+    (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) : ∃ (K : _) (_ : K ⊆ A), IsCompact K ∧ r < μ K :=
   Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
@@ -628,7 +628,7 @@ namespace WeaklyRegular
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
-    (hr : r < μ U) : ∃ (F : _)(_ : F ⊆ U), IsClosed F ∧ r < μ F :=
+    (hr : r < μ U) : ∃ (F : _) (_ : F ⊆ U), IsClosed F ∧ r < μ F :=
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 
@@ -649,14 +649,14 @@ theorem innerRegular_measurable [WeaklyRegular μ] :
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
 theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs : MeasurableSet s)
-    (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ s), IsClosed K ∧ μ s < μ K + ε :=
+    (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _) (_ : K ⊆ s), IsClosed K ∧ μ s < μ K + ε :=
   innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ (F : _)(_ : F ⊆ A), IsClosed F ∧ μ (A \ F) < ε :=
+    ∃ (F : _) (_ : F ⊆ A), IsClosed F ∧ μ (A \ F) < ε :=
   by
   rcases hA.exists_is_closed_lt_add h'A hε with ⟨F, hFA, hFc, hF⟩
   exact
@@ -670,7 +670,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
 inside by closed sets. -/
 theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) :
-    ∃ (K : _)(_ : K ⊆ A), IsClosed K ∧ r < μ K :=
+    ∃ (K : _) (_ : K ⊆ A), IsClosed K ∧ r < μ K :=
   innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 

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

@@ -137,7 +137,7 @@ proofs or statements do not apply directly.
 
 open Set Filter
 
-open ENNReal Topology NNReal BigOperators
+open scoped ENNReal Topology NNReal BigOperators
 
 namespace MeasureTheory
 

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

@@ -161,12 +161,6 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSupₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
@@ -176,12 +170,6 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
@@ -193,12 +181,6 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
     exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
 #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
 
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 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
     (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
@@ -211,12 +193,6 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
 #align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map
 
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 theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q :=
   by
   intro U hU r hr
@@ -291,12 +267,6 @@ instance zero : OuterRegular (0 : Measure α) :=
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 -/
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
@@ -309,12 +279,6 @@ theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0
   exact ⟨U, (subset_to_measurable _ _).trans hAU, hUo, hU⟩
 #align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_lt
 
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 /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets
 containing it. -/
 theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
@@ -325,24 +289,12 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
   simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
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-Case conversion may be inaccurate. Consider using '#align set.exists_is_open_le_add Set.exists_isOpen_le_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
@@ -353,12 +305,6 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -369,12 +315,6 @@ theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA :
   exact measure_diff_lt_of_lt_add hA hAU hA' hU
 #align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_lt
 
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 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β]
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular :=
   by
@@ -386,12 +326,6 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
   rwa [map_apply f.measurable this.measurable_set, f.preimage_symm, f.preimage_image]
 #align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.map
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smulₓ'. -/
 protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) :
     (x • μ).OuterRegular := by
   rcases eq_or_ne x 0 with (rfl | h0)
@@ -451,12 +385,6 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_openₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
@@ -599,12 +527,6 @@ instance zero : Regular (0 : Measure α) :=
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 -/
 
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-Case conversion may be inaccurate. Consider using '#align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -612,35 +534,17 @@ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U)
   Regular.innerRegular hU r hr
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
 
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-Case conversion may be inaccurate. Consider using '#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompactₓ'. -/
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
 theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
     μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K :=
   Regular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_nullₓ'. -/
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurableₓ'. -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -649,12 +553,6 @@ theorem innerRegular_measurable [Regular μ] :
   Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -663,12 +561,6 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
   Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
@@ -684,12 +576,6 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
@@ -698,12 +584,6 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
   Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
@@ -712,12 +592,6 @@ theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Se
   Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 
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 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
     [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).regular :=
   by
@@ -729,12 +603,6 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
         (fun K hK => hK.MeasurableSet) fun U hU => hU.MeasurableSet⟩
 #align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map
 
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 protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).regular :=
   by
   haveI := outer_regular.smul μ hx
@@ -757,12 +625,6 @@ end Regular
 
 namespace WeaklyRegular
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -770,12 +632,6 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
@@ -783,24 +639,12 @@ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : M
   WeaklyRegular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
 
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 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
@@ -809,12 +653,6 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
   innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -827,12 +665,6 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
         hF⟩
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -842,12 +674,6 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
   innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 
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-Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -856,12 +682,6 @@ theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A
   innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
 
-/- warning: measure_theory.measure.weakly_regular.restrict_of_measurable_set -> MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSetₓ'. -/
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
 theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)

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

@@ -395,8 +395,7 @@ Case conversion may be inaccurate. Consider using '#align measure_theory.measure
 protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) :
     (x • μ).OuterRegular := by
   rcases eq_or_ne x 0 with (rfl | h0)
-  · rw [zero_smul]
-    exact outer_regular.zero
+  · rw [zero_smul]; exact outer_regular.zero
   · refine' ⟨fun A hA r hr => _⟩
     rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr
     simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
@@ -431,8 +430,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     by
     intro n
     have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
-    have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
-      rw [H₁]
+    have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by rw [H₁];
       exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).Ne
     rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
     rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
@@ -467,10 +465,8 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   by
   rintro s ⟨hs, hμs⟩ r hr
-  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) :=
-    by
-    use (μ s - r) / 2
-    simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
+  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
+    use (μ s - r) / 2; simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_is_open_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
   rcases(U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
   replace hsU' := diff_subset_comm.1 hsU'
@@ -480,9 +476,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     μ s ≤ μ U := μ.mono hsU
     _ < μ K + ε := hKr
     _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
-    _ ≤ μ (K \ U') + ε + ε := by
-      mono*
-      exacts[hμU'.le, le_rfl]
+    _ ≤ μ (K \ U') + ε + ε := by mono*; exacts[hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
     
 #align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
@@ -512,8 +506,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
     rcases exists_between hr with ⟨r', hsr', hr'r⟩
     rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with ⟨-, -, U, hsU, -, hUo, -, H⟩
     refine' ⟨U, hsU, hUo, _⟩
-    rw [add_tsub_cancel_of_le hsr'.le] at H
-    exact H.trans_lt hr'r
+    rw [add_tsub_cancel_of_le hsr'.le] at H; exact H.trans_lt hr'r
   refine' MeasurableSet.induction_on_open _ _ _
   /- The proof is by measurable induction: we should check that the property is true for the empty
     set, for open sets, and is stable by taking the complement and by taking countable disjoint
@@ -531,17 +524,14 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
         hFc.is_open_compl, _⟩
     simp only [measure_compl_le_add_iff, *, hUo.measurable_set, hFc.measurable_set, true_and_iff]
   -- check for disjoint unions
-  · intro s hsd hsm H ε ε0
-    have ε0' : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
+  · intro s hsd hsm H ε ε0; have ε0' : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
     rcases ENNReal.exists_pos_sum_of_countable' ε0' ℕ with ⟨δ, δ0, hδε⟩
     choose F hFs U hsU hFc hUo hF hU using fun n => H n (δ n) (δ0 n).ne'
     -- the approximating closed set is constructed by considering finitely many sets `s i`, which
     -- cover all the measure up to `ε/2`, approximating each of these by a closed set `F i`, and
     -- taking the union of these (finitely many) `F i`.
-    have : tendsto (fun t => (∑ k in t, μ (s k)) + ε / 2) at_top (𝓝 <| μ (⋃ n, s n) + ε / 2) :=
-      by
-      rw [measure_Union hsd hsm]
-      exact tendsto.add ennreal.summable.has_sum tendsto_const_nhds
+    have : tendsto (fun t => (∑ k in t, μ (s k)) + ε / 2) at_top (𝓝 <| μ (⋃ n, s n) + ε / 2) := by
+      rw [measure_Union hsd hsm]; exact tendsto.add ennreal.summable.has_sum tendsto_const_nhds
     rcases(this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
     -- the approximating open set is constructed by taking for each `s n` an approximating open set
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
@@ -549,10 +539,8 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
       ⟨⋃ k ∈ t, F k, Union_mono fun k => Union_subset fun _ => hFs _, ⋃ n, U n, Union_mono hsU,
         isClosed_biUnion t.finite_to_set fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
     · calc
-        (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 :=
-          by
-          rw [← sum_add_distrib]
-          exact add_le_add_right (sum_le_sum fun k hk => hF k) _
+        (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 := by
+          rw [← sum_add_distrib]; exact add_le_add_right (sum_le_sum fun k hk => hF k) _
         _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
           (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _)
         _ = μ (⋃ k ∈ t, F k) + ε := _
@@ -596,8 +584,7 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
     by
     rw [← measure_Union_eq_supr, hBU]
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
-  rw [this] at hr
-  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
+  rw [this] at hr; rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
 #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
 -/
@@ -882,8 +869,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
   by
   haveI : Fact (μ A < ∞) := ⟨h'A.lt_top⟩
   refine' inner_regular.weakly_regular_of_finite _ fun V V_open => _
-  simp only [restrict_apply' hA]
-  intro r hr
+  simp only [restrict_apply' hA]; intro r hr
   have : μ (V ∩ A) ≠ ∞ := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _)
   rcases(V_open.measurable_set.inter hA).exists_lt_isClosed_of_ne_top this hr with
     ⟨F, hFVA, hFc, hF⟩

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.measure.regular
-! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
+! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.MeasureTheory.Constructions.BorelSpace.Basic
 /-!
 # Regular measures
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A measure is `outer_regular` if the measure of any measurable set `A` is the infimum of `μ U` over
 all open sets `U` containing `A`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -141,6 +141,7 @@ namespace MeasureTheory
 namespace Measure
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+#print MeasureTheory.Measure.InnerRegular /-
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
 of measure greater than `r`.
@@ -150,12 +151,19 @@ repeating the proofs. -/
 def InnerRegular {α} {m : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
   ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ (K : _)(_ : K ⊆ U), p K ∧ r < μ K
 #align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegular
+-/
 
 namespace InnerRegular
 
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
+/- warning: measure_theory.measure.inner_regular.measure_eq_supr -> MeasureTheory.Measure.InnerRegular.measure_eq_iSup is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (q U) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (p K) (fun (hK : p K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ K)))))
+but is expected to have type
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (q U) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (p K) (fun (hK : p K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) K)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSupₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
@@ -165,6 +173,12 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 
+/- warning: measure_theory.measure.inner_regular.exists_subset_lt_add -> MeasureTheory.Measure.InnerRegular.exists_subset_lt_add is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α} {ε : ENNReal}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) -> (q U) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ U) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => And (p 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.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) 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.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ K) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α} {ε : ENNReal}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) -> (q U) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) U) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (And (p 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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m 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)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) K) ε)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
@@ -176,6 +190,12 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
     exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
 #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
 
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+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] {μ : MeasureTheory.Measure.{u1} α _inst_1} {pa : (Set.{u1} α) -> Prop} {qa : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ pa qa) -> (forall (f : Equiv.{succ u1, succ u2} α β), (AEMeasurable.{u1, u2} α β _inst_2 _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) μ) -> (forall {pb : (Set.{u2} β) -> Prop} {qb : (Set.{u2} β) -> Prop}, (forall (U : Set.{u2} β), (qb U) -> (qa (Set.preimage.{u1, u2} α β (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) U))) -> (forall (K : Set.{u1} α), (pa K) -> (pb (Set.image.{u1, u2} α β (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) K))) -> (forall (K : Set.{u2} β), (pb K) -> (MeasurableSet.{u2} β _inst_2 K)) -> (forall (U : Set.{u2} β), (qb U) -> (MeasurableSet.{u2} β _inst_2 U)) -> (MeasureTheory.Measure.InnerRegular.{u2} β _inst_2 (MeasureTheory.Measure.map.{u1, u2} α β _inst_2 _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) μ) pb qb)))
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+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u1} β] {μ : MeasureTheory.Measure.{u2} α _inst_1} {pa : (Set.{u2} α) -> Prop} {qa : (Set.{u2} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u2} α _inst_1 μ pa qa) -> (forall (f : Equiv.{succ u2, succ u1} α β), (AEMeasurable.{u2, u1} α β _inst_2 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) μ) -> (forall {pb : (Set.{u1} β) -> Prop} {qb : (Set.{u1} β) -> Prop}, (forall (U : Set.{u1} β), (qb U) -> (qa (Set.preimage.{u2, u1} α β (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) U))) -> (forall (K : Set.{u2} α), (pa K) -> (pb (Set.image.{u2, u1} α β (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) K))) -> (forall (K : Set.{u1} β), (pb K) -> (MeasurableSet.{u1} β _inst_2 K)) -> (forall (U : Set.{u1} β), (qb U) -> (MeasurableSet.{u1} β _inst_2 U)) -> (MeasureTheory.Measure.InnerRegular.{u1} β _inst_2 (MeasureTheory.Measure.map.{u2, u1} α β _inst_2 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) μ) pb qb)))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.mapₓ'. -/
 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
     (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
@@ -188,6 +208,12 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
   rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
 #align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map
 
+/- warning: measure_theory.measure.inner_regular.smul -> MeasureTheory.Measure.InnerRegular.smul is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (forall (c : ENNReal), MeasureTheory.Measure.InnerRegular.{u1} α m (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) m) c μ) p q)
+but is expected to have type
+  forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (forall (c : ENNReal), MeasureTheory.Measure.InnerRegular.{u1} α m (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m)) c μ) p q)
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smulₓ'. -/
 theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q :=
   by
   intro U hU r hr
@@ -195,17 +221,20 @@ theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
 
+#print MeasureTheory.Measure.InnerRegular.trans /-
 theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
     InnerRegular μ p q' := by
   intro U hU r hr
   rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
   exact ⟨K, hKF.trans hFU, hpK, hrK⟩
 #align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegular.trans
+-/
 
 end InnerRegular
 
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
+#print MeasureTheory.Measure.OuterRegular /-
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
@@ -216,7 +245,9 @@ class OuterRegular (μ : Measure α) : Prop where
   OuterRegular :
     ∀ ⦃A : Set α⦄, MeasurableSet A → ∀ r > μ A, ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < r
 #align measure_theory.measure.outer_regular MeasureTheory.Measure.OuterRegular
+-/
 
+#print MeasureTheory.Measure.Regular /-
 /-- A measure `μ` is regular if
   - it is finite on all compact sets;
   - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable;
@@ -226,7 +257,9 @@ class OuterRegular (μ : Measure α) : Prop where
 class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
   InnerRegular : InnerRegular μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
+-/
 
+#print MeasureTheory.Measure.WeaklyRegular /-
 /-- A measure `μ` is weakly regular if
   - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable;
   - it is inner regular for open sets, using closed sets:
@@ -235,7 +268,9 @@ class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular
 class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
   InnerRegular : InnerRegular μ IsClosed IsOpen
 #align measure_theory.measure.weakly_regular MeasureTheory.Measure.WeaklyRegular
+-/
 
+#print MeasureTheory.Measure.Regular.weaklyRegular /-
 -- see Note [lower instance priority]
 /-- A regular measure is weakly regular. -/
 instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : WeaklyRegular μ
@@ -243,13 +278,22 @@ instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : Wea
     let ⟨K, hKU, hcK, hK⟩ := Regular.innerRegular hU r hr
     ⟨K, hKU, hcK.IsClosed, hK⟩
 #align measure_theory.measure.regular.weakly_regular MeasureTheory.Measure.Regular.weaklyRegular
+-/
 
 namespace OuterRegular
 
+#print MeasureTheory.Measure.OuterRegular.zero /-
 instance zero : OuterRegular (0 : Measure α) :=
   ⟨fun A hA r hr => ⟨univ, subset_univ A, isOpen_univ, hr⟩⟩
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
+-/
 
+/- warning: set.exists_is_open_lt_of_lt -> Set.exists_isOpen_lt_of_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α) (r : ENNReal), (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) r) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 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))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) r))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α) (r : ENNReal), (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) r) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 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))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) r))))
+Case conversion may be inaccurate. Consider using '#align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
@@ -262,6 +306,12 @@ theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0
   exact ⟨U, (subset_to_measurable _ _).trans hAU, hUo, hU⟩
 #align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_lt
 
+/- warning: set.measure_eq_infi_is_open -> Set.measure_eq_iInf_isOpen is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpenₓ'. -/
 /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets
 containing it. -/
 theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
@@ -272,12 +322,24 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular
   simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
 #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 
+/- warning: set.exists_is_open_lt_add -> Set.exists_isOpen_lt_add is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 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))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 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))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _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)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) ε))))))
+Case conversion may be inaccurate. Consider using '#align set.exists_is_open_lt_add Set.exists_isOpen_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
+/- warning: set.exists_is_open_le_add -> Set.exists_isOpen_le_add is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _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)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) ε)))))
+Case conversion may be inaccurate. Consider using '#align set.exists_is_open_le_add Set.exists_isOpen_le_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
@@ -288,6 +350,12 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 
+/- warning: measurable_set.exists_is_open_diff_lt -> MeasurableSet.exists_isOpen_diff_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {A : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 U) (And (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) U A)) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {A : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 U) (And (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) U A)) ε))))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -298,6 +366,12 @@ theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA :
   exact measure_diff_lt_of_lt_add hA hAU hA' hU
 #align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_lt
 
+/- warning: measure_theory.measure.outer_regular.map -> MeasureTheory.Measure.OuterRegular.map is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : BorelSpace.{u2} β _inst_5 _inst_4] (f : Homeomorph.{u1, u2} α β _inst_2 _inst_5) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_7 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.OuterRegular.{u2} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u1, u2} α β _inst_4 _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Homeomorph.{u1, u2} α β _inst_2 _inst_5) (fun (_x : Homeomorph.{u1, u2} α β _inst_2 _inst_5) => α -> β) (Homeomorph.hasCoeToFun.{u1, u2} α β _inst_2 _inst_5) f) μ)
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OpensMeasurableSpace.{u2} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : BorelSpace.{u1} β _inst_5 _inst_4] (f : Homeomorph.{u2, u1} α β _inst_2 _inst_5) (μ : MeasureTheory.Measure.{u2} α _inst_1) [_inst_7 : MeasureTheory.Measure.OuterRegular.{u2} α _inst_1 _inst_2 μ], MeasureTheory.Measure.OuterRegular.{u1} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u2, u1} α β _inst_4 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α (fun (_x : α) => β) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (Homeomorph.instEquivLikeHomeomorph.{u2, u1} α β _inst_2 _inst_5))) f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.mapₓ'. -/
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β]
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular :=
   by
@@ -309,6 +383,12 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
   rwa [map_apply f.measurable this.measurable_set, f.preimage_symm, f.preimage_image]
 #align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.map
 
+/- warning: measure_theory.measure.outer_regular.smul -> MeasureTheory.Measure.OuterRegular.smul is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) _inst_1) x μ))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.{u1} α _inst_1) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) _inst_1)) x μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smulₓ'. -/
 protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) :
     (x • μ).OuterRegular := by
   rcases eq_or_ne x 0 with (rfl | h0)
@@ -322,6 +402,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 end OuterRegular
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+#print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /-
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
 protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
@@ -363,11 +444,18 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     _ < r := hδε
     
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
+-/
 
 namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
+/- warning: measure_theory.measure.inner_regular.measurable_set_of_open -> MeasureTheory.Measure.InnerRegular.measurableSet_of_open is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} {p : (Set.{u1} α) -> Prop} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (IsOpen.{u1} α _inst_2)) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) -> (forall {{s : Set.{u1} α}} {{U : Set.{u1} α}}, (p s) -> (IsOpen.{u1} α _inst_2 U) -> (p (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s U))) -> (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} {p : (Set.{u1} α) -> Prop} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (IsOpen.{u1} α _inst_2)) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) -> (forall {{s : Set.{u1} α}} {{U : Set.{u1} α}}, (p s) -> (IsOpen.{u1} α _inst_2 U) -> (p (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s U))) -> (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_openₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
@@ -401,6 +489,7 @@ open Finset
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+#print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /-
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
 theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure μ]
@@ -475,7 +564,9 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
         
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite
+-/
 
+#print MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace /-
 /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
 by closed sets. -/
 theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
@@ -487,7 +578,9 @@ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpa
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
 #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
+-/
 
+#print MeasureTheory.Measure.InnerRegular.isCompact_isClosed /-
 /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/
 theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X]
     [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed :=
@@ -504,15 +597,24 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
 #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
+-/
 
 end InnerRegular
 
 namespace Regular
 
+#print MeasureTheory.Measure.Regular.zero /-
 instance zero : Regular (0 : Measure α) :=
   ⟨fun U hU r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
+-/
 
+/- warning: is_open.exists_lt_is_compact -> IsOpen.exists_lt_isCompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => And (IsCompact.{u1} α _inst_2 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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (And (IsCompact.{u1} α _inst_2 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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))))
+Case conversion may be inaccurate. Consider using '#align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -520,17 +622,35 @@ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U)
   Regular.innerRegular hU r hr
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
 
+/- warning: is_open.measure_eq_supr_is_compact -> IsOpen.measure_eq_iSup_isCompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsCompact.{u1} α _inst_2 K) (fun (h2 : IsCompact.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsCompact.{u1} α _inst_2 K) (fun (h2 : IsCompact.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))
+Case conversion may be inaccurate. Consider using '#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompactₓ'. -/
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
 theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
     μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K :=
   Regular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 
+/- warning: measure_theory.measure.regular.exists_compact_not_null -> MeasureTheory.Measure.Regular.exists_compact_not_null is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Iff (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (IsCompact.{u1} α _inst_2 K) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))))) (Ne.{succ u1} (MeasureTheory.Measure.{u1} α _inst_1) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Iff (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (IsCompact.{u1} α _inst_2 K) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))))) (Ne.{succ u1} (MeasureTheory.Measure.{u1} α _inst_1) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_nullₓ'. -/
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 
+/- warning: measure_theory.measure.regular.inner_regular_measurable -> MeasureTheory.Measure.Regular.innerRegular_measurable is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsCompact.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsCompact.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurableₓ'. -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -539,6 +659,12 @@ theorem innerRegular_measurable [Regular μ] :
   Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 
+/- warning: measurable_set.exists_is_compact_lt_add -> MeasurableSet.exists_isCompact_lt_add is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) ε))))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -547,6 +673,12 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
   Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
+/- warning: measurable_set.exists_is_compact_diff_lt -> MeasurableSet.exists_isCompact_diff_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : T2Space.{u1} α _inst_2] [_inst_5 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A K)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : T2Space.{u1} α _inst_2] [_inst_5 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) A K)) ε)))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
@@ -562,6 +694,12 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 
+/- warning: measurable_set.exists_lt_is_compact_of_ne_top -> MeasurableSet.exists_lt_isCompact_of_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {r : ENNReal}, (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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {r : ENNReal}, (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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
@@ -570,6 +708,12 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
   Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
+/- warning: measurable_set.measure_eq_supr_is_compact_of_ne_top -> MeasurableSet.measure_eq_iSup_isCompact_of_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsCompact.{u1} α _inst_2 K) (fun (h : IsCompact.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsCompact.{u1} α _inst_2 K) (fun (h : IsCompact.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
@@ -578,6 +722,12 @@ theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Se
   Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 
+/- warning: measure_theory.measure.regular.map -> MeasureTheory.Measure.Regular.map is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : T2Space.{u2} β _inst_5] [_inst_7 : BorelSpace.{u2} β _inst_5 _inst_4] [_inst_8 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] (f : Homeomorph.{u1, u2} α β _inst_2 _inst_5), MeasureTheory.Measure.Regular.{u2} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u1, u2} α β _inst_4 _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Homeomorph.{u1, u2} α β _inst_2 _inst_5) (fun (_x : Homeomorph.{u1, u2} α β _inst_2 _inst_5) => α -> β) (Homeomorph.hasCoeToFun.{u1, u2} α β _inst_2 _inst_5) f) μ)
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] {μ : MeasureTheory.Measure.{u2} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u2} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : T2Space.{u1} β _inst_5] [_inst_7 : BorelSpace.{u1} β _inst_5 _inst_4] [_inst_8 : MeasureTheory.Measure.Regular.{u2} α _inst_1 _inst_2 μ] (f : Homeomorph.{u2, u1} α β _inst_2 _inst_5), MeasureTheory.Measure.Regular.{u1} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u2, u1} α β _inst_4 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α (fun (_x : α) => β) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (Homeomorph.instEquivLikeHomeomorph.{u2, u1} α β _inst_2 _inst_5))) f) μ)
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.mapₓ'. -/
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
     [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).regular :=
   by
@@ -589,6 +739,12 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
         (fun K hK => hK.MeasurableSet) fun U hU => hU.MeasurableSet⟩
 #align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map
 
+/- warning: measure_theory.measure.regular.smul -> MeasureTheory.Measure.Regular.smul is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) _inst_1) x μ))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.{u1} α _inst_1) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) _inst_1)) x μ))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smulₓ'. -/
 protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).regular :=
   by
   haveI := outer_regular.smul μ hx
@@ -596,6 +752,7 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • 
   exact ⟨regular.inner_regular.smul x⟩
 #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
 
+#print MeasureTheory.Measure.Regular.sigmaFinite /-
 -- see Note [lower instance priority]
 /-- A regular measure in a σ-compact space is σ-finite. -/
 instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : SigmaFinite μ :=
@@ -604,11 +761,18 @@ instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : Sig
         Finite := fun n => (isCompact_compactCovering α n).measure_lt_top
         spanning := iUnion_compactCovering α }⟩⟩
 #align measure_theory.measure.regular.sigma_finite MeasureTheory.Measure.Regular.sigmaFinite
+-/
 
 end Regular
 
 namespace WeaklyRegular
 
+/- warning: is_open.exists_lt_is_closed -> IsOpen.exists_lt_isClosed is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) => And (IsClosed.{u1} α _inst_2 F) (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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ F))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) (And (IsClosed.{u1} α _inst_2 F) (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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) F))))))
+Case conversion may be inaccurate. Consider using '#align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
@@ -616,6 +780,12 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 
+/- warning: is_open.measure_eq_supr_is_closed -> IsOpen.measure_eq_iSup_isClosed is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (F : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsClosed.{u1} α _inst_2 F) (fun (h : IsClosed.{u1} α _inst_2 F) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ F)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (F : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsClosed.{u1} α _inst_2 F) (fun (h : IsClosed.{u1} α _inst_2 F) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) F)))))
+Case conversion may be inaccurate. Consider using '#align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
@@ -623,12 +793,24 @@ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : M
   WeaklyRegular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
 
+/- warning: measure_theory.measure.weakly_regular.inner_regular_measurable -> MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsClosed.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsClosed.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurableₓ'. -/
 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
+/- warning: measurable_set.exists_is_closed_lt_add -> MeasurableSet.exists_isClosed_lt_add is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K s) => And (IsClosed.{u1} α _inst_2 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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (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.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K s) (And (IsClosed.{u1} α _inst_2 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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (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} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) ε))))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_addₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
@@ -637,6 +819,12 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
   innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
+/- warning: measurable_set.exists_is_closed_diff_lt -> MeasurableSet.exists_isClosed_diff_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F A) => And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A F)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F A) (And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) A F)) ε)))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_ltₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
@@ -649,6 +837,12 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
         hF⟩
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 
+/- warning: measurable_set.exists_lt_is_closed_of_ne_top -> MeasurableSet.exists_lt_isClosed_of_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {r : ENNReal}, (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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsClosed.{u1} α _inst_2 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))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {r : ENNReal}, (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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsClosed.{u1} α _inst_2 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))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -658,6 +852,12 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
   innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 
+/- warning: measurable_set.measure_eq_supr_is_closed_of_ne_top -> MeasurableSet.measure_eq_iSup_isClosed_of_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsClosed.{u1} α _inst_2 K) (fun (h : IsClosed.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsClosed.{u1} α _inst_2 K) (fun (h : IsClosed.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))
+Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_topₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
@@ -666,6 +866,12 @@ theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A
   innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
 
+/- warning: measure_theory.measure.weakly_regular.restrict_of_measurable_set -> MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : BorelSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 (MeasureTheory.Measure.restrict.{u1} α _inst_1 μ A))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : BorelSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 (MeasureTheory.Measure.restrict.{u1} α _inst_1 μ A))
+Case conversion may be inaccurate. Consider using '#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSetₓ'. -/
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
 theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
@@ -682,6 +888,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
 #align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
 
+#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure /-
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
 instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
@@ -689,11 +896,13 @@ instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
     WeaklyRegular μ :=
   (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
 #align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure
+-/
 
+#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite /-
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
-instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite {X : Type _}
+instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [LocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
@@ -702,12 +911,14 @@ instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite
     have : Fact (μ U < ∞) := ⟨hU.2⟩
     exact ⟨hU.1, inferInstance⟩
   ⟨inner_regular.of_pseudo_emetric_space μ⟩
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudo_emetric_second_countable_of_locally_finite
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite
+-/
 
 end WeaklyRegular
 
 attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
+#print MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure /-
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
 instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure {X : Type _}
@@ -717,6 +928,7 @@ instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure
   lt_top_of_isCompact K hK := hK.measure_lt_top
   InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
 #align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure
+-/
 
 end Measure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.measure.regular
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
+! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.MeasureTheory.Constructions.BorelSpace
+import Mathbin.MeasureTheory.Constructions.BorelSpace.Basic
 
 /-!
 # Regular measures

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

@@ -157,13 +157,13 @@ variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α
   {ε : ℝ≥0∞}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
-theorem measure_eq_supᵢ (H : InnerRegular μ p q) (hU : q U) :
+theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
   by
   refine'
-    le_antisymm (le_of_forall_lt fun r hr => _) (supᵢ₂_le fun K hK => supᵢ_le fun _ => μ.mono hK)
-  simpa only [lt_supᵢ_iff, exists_prop] using H hU r hr
-#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_supᵢ
+    le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
+  simpa only [lt_iSup_iff, exists_prop] using H hU r hr
+#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
@@ -192,7 +192,7 @@ theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ
   by
   intro U hU r hr
   rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr
-  simpa only [ENNReal.mul_supᵢ, lt_supᵢ_iff, exists_prop] using hr
+  simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
 
 theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
@@ -264,13 +264,13 @@ theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0
 
 /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets
 containing it. -/
-theorem Set.measure_eq_infᵢ_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
+theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
     μ A = ⨅ (U : Set α) (h : A ⊆ U) (h2 : IsOpen U), μ U :=
   by
-  refine' le_antisymm (le_infᵢ₂ fun s hs => le_infᵢ fun h2s => μ.mono hs) _
+  refine' le_antisymm (le_iInf₂ fun s hs => le_iInf fun h2s => μ.mono hs) _
   refine' le_of_forall_lt' fun r hr => _
-  simpa only [infᵢ_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
-#align set.measure_eq_infi_is_open Set.measure_eq_infᵢ_isOpen
+  simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
+#align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
@@ -316,7 +316,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
     exact outer_regular.zero
   · refine' ⟨fun A hA r hr => _⟩
     rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr
-    simpa only [ENNReal.mul_infᵢ_of_ne h0 hx, gt_iff_lt, infᵢ_lt_iff, exists_prop] using hr
+    simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
 
 end OuterRegular
@@ -340,7 +340,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
       ⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
         (inter_subset_right _ _).trans (disjointed_subset _ _),
         (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
-    rw [← inter_Union, unionᵢ_disjointed, s.spanning, inter_univ]
+    rw [← inter_Union, iUnion_disjointed, s.spanning, inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
   rw [lt_tsub_iff_right, add_comm] at hδε
   have : ∀ n, ∃ (U : _)(_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n :=
@@ -354,7 +354,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
     exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
   choose U hAU hUo hU
-  refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_unionᵢ hUo, _⟩
+  refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_iUnion hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
     _ ≤ ∑' n, μ (A n) + δ n := (ENNReal.tsum_le_tsum fun n => (hU n).le)
@@ -455,7 +455,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
     refine'
       ⟨⋃ k ∈ t, F k, Union_mono fun k => Union_subset fun _ => hFs _, ⋃ n, U n, Union_mono hsU,
-        isClosed_bunionᵢ t.finite_to_set fun k _ => hFc k, isOpen_unionᵢ hUo, ht.le.trans _, _⟩
+        isClosed_biUnion t.finite_to_set fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
     · calc
         (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 :=
           by
@@ -484,7 +484,7 @@ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpa
   intro U hU r hr
   rcases hU.exists_Union_is_closed with ⟨F, F_closed, -, rfl, F_mono⟩
   rw [measure_Union_eq_supr F_mono.directed_le] at hr
-  rcases lt_supᵢ_iff.1 hr with ⟨n, hn⟩
+  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
 #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
 
@@ -495,13 +495,13 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
   intro F hF r hr
   set B : ℕ → Set X := compactCovering X
   have hBc : ∀ n, IsCompact (F ∩ B n) := fun n => (isCompact_compactCovering X n).inter_left hF
-  have hBU : (⋃ n, F ∩ B n) = F := by rw [← inter_Union, unionᵢ_compactCovering, Set.inter_univ]
+  have hBU : (⋃ n, F ∩ B n) = F := by rw [← inter_Union, iUnion_compactCovering, Set.inter_univ]
   have : μ F = ⨆ n, μ (F ∩ B n) :=
     by
     rw [← measure_Union_eq_supr, hBU]
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
   rw [this] at hr
-  rcases lt_supᵢ_iff.1 hr with ⟨n, hn⟩
+  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
 #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
 
@@ -521,14 +521,14 @@ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U)
 #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
 
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
-theorem IsOpen.measure_eq_supᵢ_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
+theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] :
     μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K :=
-  Regular.innerRegular.measure_eq_supᵢ hU
-#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_supᵢ_isCompact
+  Regular.innerRegular.measure_eq_iSup hU
+#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
-    ENNReal.supᵢ_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
@@ -573,10 +573,10 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
-theorem MeasurableSet.measure_eq_supᵢ_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
+theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsCompact K), μ K :=
-  Regular.innerRegular_measurable.measure_eq_supᵢ ⟨hA, h'A⟩
-#align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_supᵢ_isCompact_of_ne_top
+  Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
+#align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
     [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).regular :=
@@ -602,7 +602,7 @@ instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : Sig
   ⟨⟨{   Set := compactCovering α
         set_mem := fun n => trivial
         Finite := fun n => (isCompact_compactCovering α n).measure_lt_top
-        spanning := unionᵢ_compactCovering α }⟩⟩
+        spanning := iUnion_compactCovering α }⟩⟩
 #align measure_theory.measure.regular.sigma_finite MeasureTheory.Measure.Regular.sigmaFinite
 
 end Regular
@@ -618,10 +618,10 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
-theorem IsOpen.measure_eq_supᵢ_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
+theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
     [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (h : IsClosed F), μ F :=
-  WeaklyRegular.innerRegular.measure_eq_supᵢ hU
-#align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_supᵢ_isClosed
+  WeaklyRegular.innerRegular.measure_eq_iSup hU
+#align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
 
 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
@@ -661,10 +661,10 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
-theorem MeasurableSet.measure_eq_supᵢ_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
+theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsClosed K), μ K :=
-  innerRegular_measurable.measure_eq_supᵢ ⟨hA, h'A⟩
-#align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_supᵢ_isClosed_of_ne_top
+  innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
+#align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
 
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -223,7 +223,7 @@ class OuterRegular (μ : Measure α) : Prop where
   - it is inner regular for open sets, using compact sets:
     `μ(U) = sup {μ(K) | K ⊆ U compact}` for `U` open. -/
 @[protect_proj]
-class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
+class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
   InnerRegular : InnerRegular μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
 
@@ -403,7 +403,7 @@ open Finset
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
-theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
+theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure μ]
     (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ :=
   by
   have hfin : ∀ {s}, μ s ≠ ⊤ := measure_ne_top μ
@@ -684,18 +684,18 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
 
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
-    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
+instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
+    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [FiniteMeasure μ] :
     WeaklyRegular μ :=
   (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
 instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite {X : Type _}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
-    [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
+    [BorelSpace X] (μ : Measure X) [LocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
     by
     refine' (μ.finite_spanning_sets_in_open'.mono' fun U hU => _).OuterRegular
@@ -710,13 +710,13 @@ attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
-instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type _}
+instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure {X : Type _}
     [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
-    [IsLocallyFiniteMeasure μ] : Regular μ
+    [LocallyFiniteMeasure μ] : Regular μ
     where
   lt_top_of_isCompact K hK := hK.measure_lt_top
   InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
-#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
+#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure
 
 end Measure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -177,7 +177,7 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
 #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
 
 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
-    (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AeMeasurable f μ) {pb qb : Set β → Prop}
+    (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
     (hB₁ : ∀ K, pb K → MeasurableSet K) (hB₂ : ∀ U, qb U → MeasurableSet U) :
     InnerRegular (map f μ) pb qb := by
@@ -371,7 +371,7 @@ variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
-theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
+theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
     InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
   by
@@ -394,7 +394,7 @@ theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0
       exacts[hμU'.le, le_rfl]
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
     
-#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSetOfOpen
+#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
 
 open Finset
 
@@ -403,7 +403,7 @@ open Finset
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
-theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
+theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
     (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ :=
   by
   have hfin : ∀ {s}, μ s ≠ ⊤ := measure_ne_top μ
@@ -474,11 +474,11 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
         
-#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegularOfFinite
+#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite
 
 /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
 by closed sets. -/
-theorem ofPseudoEmetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
+theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
     (μ : Measure X) : InnerRegular μ IsClosed IsOpen :=
   by
   intro U hU r hr
@@ -486,10 +486,10 @@ theorem ofPseudoEmetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpac
   rw [measure_Union_eq_supr F_mono.directed_le] at hr
   rcases lt_supᵢ_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_Union _ _, F_closed n, hn⟩
-#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.ofPseudoEmetricSpace
+#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
 
 /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/
-theorem isCompactIsClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X]
+theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X]
     [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed :=
   by
   intro F hF r hr
@@ -503,7 +503,7 @@ theorem isCompactIsClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X
   rw [this] at hr
   rcases lt_supᵢ_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
-#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompactIsClosed
+#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
 
 end InnerRegular
 
@@ -534,17 +534,17 @@ theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
-theorem innerRegularMeasurable [Regular μ] :
+theorem innerRegular_measurable [Regular μ] :
     InnerRegular μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  Regular.innerRegular.measurableSetOfOpen isCompact_empty fun _ _ => IsCompact.diff
-#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegularMeasurable
+  Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
+#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ A), IsCompact K ∧ μ A < μ K + ε :=
-  Regular.innerRegularMeasurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
+  Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
@@ -567,7 +567,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
     (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) : ∃ (K : _)(_ : K ⊆ A), IsCompact K ∧ r < μ K :=
-  Regular.innerRegularMeasurable ⟨hA, h'A⟩ _ hr
+  Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
@@ -575,7 +575,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
 inside by compact sets. -/
 theorem MeasurableSet.measure_eq_supᵢ_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsCompact K), μ K :=
-  Regular.innerRegularMeasurable.measure_eq_supᵢ ⟨hA, h'A⟩
+  Regular.innerRegular_measurable.measure_eq_supᵢ ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_supᵢ_isCompact_of_ne_top
 
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
@@ -623,18 +623,18 @@ theorem IsOpen.measure_eq_supᵢ_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ :
   WeaklyRegular.innerRegular.measure_eq_supᵢ hU
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_supᵢ_isClosed
 
-theorem innerRegularMeasurable [WeaklyRegular μ] :
+theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  WeaklyRegular.innerRegular.measurableSetOfOpen isClosed_empty fun _ _ h₁ h₂ =>
+  WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
     h₁.inter h₂.isClosed_compl
-#align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegularMeasurable
+#align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
 theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs : MeasurableSet s)
     (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ s), IsClosed K ∧ μ s < μ K + ε :=
-  innerRegularMeasurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
+  innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
@@ -655,7 +655,7 @@ inside by closed sets. -/
 theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) :
     ∃ (K : _)(_ : K ⊆ A), IsClosed K ∧ r < μ K :=
-  innerRegularMeasurable ⟨hA, h'A⟩ _ hr
+  innerRegular_measurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
@@ -663,13 +663,13 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
 inside by closed sets. -/
 theorem MeasurableSet.measure_eq_supᵢ_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (h : IsClosed K), μ K :=
-  innerRegularMeasurable.measure_eq_supᵢ ⟨hA, h'A⟩
+  innerRegular_measurable.measure_eq_supᵢ ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_supᵢ_isClosed_of_ne_top
 
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
-theorem restrictOfMeasurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α) (hA : MeasurableSet A)
-    (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) :=
+theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
+    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) :=
   by
   haveI : Fact (μ A < ∞) := ⟨h'A.lt_top⟩
   refine' inner_regular.weakly_regular_of_finite _ fun V V_open => _
@@ -680,19 +680,20 @@ theorem restrictOfMeasurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
     ⟨F, hFVA, hFc, hF⟩
   refine' ⟨F, hFVA.trans (inter_subset_left _ _), hFc, _⟩
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
-#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrictOfMeasurableSet
+#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
 
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) ofPseudoEmetricSpaceOfIsFiniteMeasure {X : Type _} [PseudoEMetricSpace X]
-    [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] : WeaklyRegular μ :=
-  (InnerRegular.ofPseudoEmetricSpace μ).weaklyRegularOfFinite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.ofPseudoEmetricSpaceOfIsFiniteMeasure
+instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
+    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
+    WeaklyRegular μ :=
+  (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
-instance (priority := 100) ofPseudoEmetricSecondCountableOfLocallyFinite {X : Type _}
+instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite {X : Type _}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
@@ -701,7 +702,7 @@ instance (priority := 100) ofPseudoEmetricSecondCountableOfLocallyFinite {X : Ty
     have : Fact (μ U < ∞) := ⟨hU.2⟩
     exact ⟨hU.1, inferInstance⟩
   ⟨inner_regular.of_pseudo_emetric_space μ⟩
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.ofPseudoEmetricSecondCountableOfLocallyFinite
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudo_emetric_second_countable_of_locally_finite
 
 end WeaklyRegular
 
@@ -709,13 +710,13 @@ attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
-instance (priority := 100) Regular.ofSigmaCompactSpaceOfIsLocallyFiniteMeasure {X : Type _}
+instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type _}
     [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
     [IsLocallyFiniteMeasure μ] : Regular μ
     where
   lt_top_of_isCompact K hK := hK.measure_lt_top
-  InnerRegular := (InnerRegular.isCompactIsClosed μ).trans (InnerRegular.ofPseudoEmetricSpace μ)
-#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.ofSigmaCompactSpaceOfIsLocallyFiniteMeasure
+  InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
+#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
 
 end Measure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -478,7 +478,7 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
 
 /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
 by closed sets. -/
-theorem ofPseudoEmetricSpace {X : Type _} [PseudoEmetricSpace X] [MeasurableSpace X]
+theorem ofPseudoEmetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
     (μ : Measure X) : InnerRegular μ IsClosed IsOpen :=
   by
   intro U hU r hr
@@ -684,7 +684,7 @@ theorem restrictOfMeasurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
 
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) ofPseudoEmetricSpaceOfIsFiniteMeasure {X : Type _} [PseudoEmetricSpace X]
+instance (priority := 100) ofPseudoEmetricSpaceOfIsFiniteMeasure {X : Type _} [PseudoEMetricSpace X]
     [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] : WeaklyRegular μ :=
   (InnerRegular.ofPseudoEmetricSpace μ).weaklyRegularOfFinite μ
 #align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.ofPseudoEmetricSpaceOfIsFiniteMeasure
@@ -693,7 +693,7 @@ instance (priority := 100) ofPseudoEmetricSpaceOfIsFiniteMeasure {X : Type _} [P
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
 instance (priority := 100) ofPseudoEmetricSecondCountableOfLocallyFinite {X : Type _}
-    [PseudoEmetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
+    [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : outer_regular μ :=
     by
@@ -705,12 +705,12 @@ instance (priority := 100) ofPseudoEmetricSecondCountableOfLocallyFinite {X : Ty
 
 end WeaklyRegular
 
-attribute [local instance] Emetric.second_countable_of_sigma_compact
+attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
 instance (priority := 100) Regular.ofSigmaCompactSpaceOfIsLocallyFiniteMeasure {X : Type _}
-    [EmetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
+    [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
     [IsLocallyFiniteMeasure μ] : Regular μ
     where
   lt_top_of_isCompact K hK := hK.measure_lt_top

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -140,7 +140,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
 of measure greater than `r`.
@@ -156,7 +156,7 @@ namespace InnerRegular
 variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem measure_eq_supᵢ (H : InnerRegular μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K :=
   by
@@ -165,7 +165,7 @@ theorem measure_eq_supᵢ (H : InnerRegular μ p q) (hU : q U) :
   simpa only [lt_supᵢ_iff, exists_prop] using H hU r hr
 #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_supᵢ
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ U), p K ∧ μ U < μ K + ε :=
   by
@@ -206,7 +206,7 @@ end InnerRegular
 
 variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
 This definition implies the same equality for any (not necessarily measurable) set, see
@@ -250,7 +250,7 @@ instance zero : OuterRegular (0 : Measure α) :=
   ⟨fun A hA r hr => ⟨univ, subset_univ A, isOpen_univ, hr⟩⟩
 #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
 measure less than `r`. -/
 theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r) :
@@ -272,13 +272,13 @@ theorem Set.measure_eq_infᵢ_isOpen (A : Set α) (μ : Measure α) [OuterRegula
   simpa only [infᵢ_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr
 #align set.measure_eq_infi_is_open Set.measure_eq_infᵢ_isOpen
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
   A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
   by
@@ -288,7 +288,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A)
     (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε :=
@@ -321,7 +321,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
 
 end OuterRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
 is outer regular, then the original measure is outer regular as well. -/
 protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
@@ -357,9 +357,9 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
   refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_unionᵢ hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-    _ ≤ ∑' n, μ (A n) + δ n := ENNReal.tsum_le_tsum fun n => (hU n).le
+    _ ≤ ∑' n, μ (A n) + δ n := (ENNReal.tsum_le_tsum fun n => (hU n).le)
     _ = (∑' n, μ (A n)) + ∑' n, δ n := ENNReal.tsum_add
-    _ = μ (⋃ n, A n) + ∑' n, δ n := congr_arg₂ (· + ·) (measure_Union hAd hAm).symm rfl
+    _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_Union hAd hAm).symm rfl)
     _ < r := hδε
     
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
@@ -368,7 +368,7 @@ namespace InnerRegular
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
 finite measure can by approximated by a (closed or compact) subset. -/
 theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
@@ -388,7 +388,7 @@ theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0
   calc
     μ s ≤ μ U := μ.mono hsU
     _ < μ K + ε := hKr
-    _ ≤ μ (K \ U') + μ U' + ε := add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _
+    _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
     _ ≤ μ (K \ U') + ε + ε := by
       mono*
       exacts[hμU'.le, le_rfl]
@@ -398,9 +398,9 @@ theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0
 
 open Finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
 theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
@@ -462,7 +462,7 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
           rw [← sum_add_distrib]
           exact add_le_add_right (sum_le_sum fun k hk => hF k) _
         _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
-          add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _
+          (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _)
         _ = μ (⋃ k ∈ t, F k) + ε := _
         
       rw [measure_bUnion_finset, add_assoc, ENNReal.add_halves]
@@ -470,7 +470,7 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
     ·
       calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-        _ ≤ ∑' n, μ (s n) + δ n := ENNReal.tsum_le_tsum hU
+        _ ≤ ∑' n, μ (s n) + δ n := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
         
@@ -513,7 +513,7 @@ instance zero : Regular (0 : Measure α) :=
   ⟨fun U hU r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩
 #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/
 /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
 theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (K : _)(_ : K ⊆ U), IsCompact K ∧ r < μ K :=
@@ -539,7 +539,7 @@ theorem innerRegularMeasurable [Regular μ] :
   Regular.innerRegular.measurableSetOfOpen isCompact_empty fun _ _ => IsCompact.diff
 #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegularMeasurable
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/
 theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
@@ -547,7 +547,7 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA
   Regular.innerRegularMeasurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add` and
 `measurable_set.exists_lt_is_compact_of_ne_top`. -/
@@ -562,7 +562,7 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/
 theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
@@ -570,7 +570,7 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α
   Regular.innerRegularMeasurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
 theorem MeasurableSet.measure_eq_supᵢ_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
@@ -609,14 +609,14 @@ end Regular
 
 namespace WeaklyRegular
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ (F : _)(_ : F ⊆ U), IsClosed F ∧ r < μ F :=
   WeaklyRegular.innerRegular hU r hr
 #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (F «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem IsOpen.measure_eq_supᵢ_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
     [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (h : IsClosed F), μ F :=
@@ -629,7 +629,7 @@ theorem innerRegularMeasurable [WeaklyRegular μ] :
     h₁.inter h₂.isClosed_compl
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegularMeasurable
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
 number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/
 theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs : MeasurableSet s)
@@ -637,7 +637,7 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs
   innerRegularMeasurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε
 #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (F «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/
 theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ (F : _)(_ : F ⊆ A), IsClosed F ∧ μ (A \ F) < ε :=
@@ -649,7 +649,7 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR
         hF⟩
 #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
@@ -658,7 +658,7 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set
   innerRegularMeasurable ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (K «expr ⊆ » A) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem MeasurableSet.measure_eq_supᵢ_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.measure.regular
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -282,11 +282,9 @@ theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ 
 theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε :=
   by
-  rcases le_or_lt ∞ (μ A) with (H | H)
-  ·
-    exact
-      ⟨univ, subset_univ _, isOpen_univ, by simp only [top_le_iff.mp H, ENNReal.top_add, le_top]⟩
-  · rcases A.exists_is_open_lt_add H.ne hε with ⟨U, AU, U_open, hU⟩
+  rcases eq_or_ne (μ A) ∞ with (H | H)
+  · exact ⟨univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]⟩
+  · rcases A.exists_is_open_lt_add H hε with ⟨U, AU, U_open, hU⟩
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
 

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

@@ -134,7 +134,7 @@ proofs or statements do not apply directly.
 
 open Set Filter
 
-open Ennreal Topology NNReal BigOperators
+open ENNReal Topology NNReal BigOperators
 
 namespace MeasureTheory
 
@@ -172,8 +172,8 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (
   cases' eq_or_ne (μ U) 0 with h₀ h₀
   · refine' ⟨∅, empty_subset _, h0, _⟩
     rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
-  · rcases H hU _ (Ennreal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
-    exact ⟨K, hKU, hKc, Ennreal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
+  · rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
+    exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
 #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
 
 theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
@@ -192,7 +192,7 @@ theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ
   by
   intro U hU r hr
   rw [smul_apply, H.measure_eq_supr hU, smul_eq_mul] at hr
-  simpa only [Ennreal.mul_supᵢ, lt_supᵢ_iff, exists_prop] using hr
+  simpa only [ENNReal.mul_supᵢ, lt_supᵢ_iff, exists_prop] using hr
 #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
 
 theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
@@ -275,7 +275,7 @@ theorem Set.measure_eq_infᵢ_isOpen (A : Set α) (μ : Measure α) [OuterRegula
 /- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
 theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
     (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε :=
-  A.exists_isOpen_lt_of_lt _ (Ennreal.lt_add_right hA hε)
+  A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
 #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U «expr ⊇ » A) -/
@@ -285,7 +285,7 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ
   rcases le_or_lt ∞ (μ A) with (H | H)
   ·
     exact
-      ⟨univ, subset_univ _, isOpen_univ, by simp only [top_le_iff.mp H, Ennreal.top_add, le_top]⟩
+      ⟨univ, subset_univ _, isOpen_univ, by simp only [top_le_iff.mp H, ENNReal.top_add, le_top]⟩
   · rcases A.exists_is_open_lt_add H.ne hε with ⟨U, AU, U_open, hU⟩
     exact ⟨U, AU, U_open, hU.le⟩
 #align set.exists_is_open_le_add Set.exists_isOpen_le_add
@@ -318,7 +318,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
     exact outer_regular.zero
   · refine' ⟨fun A hA r hr => _⟩
     rw [smul_apply, A.measure_eq_infi_is_open, smul_eq_mul] at hr
-    simpa only [Ennreal.mul_infᵢ_of_ne h0 hx, gt_iff_lt, infᵢ_lt_iff, exists_prop] using hr
+    simpa only [ENNReal.mul_infᵢ_of_ne h0 hx, gt_iff_lt, infᵢ_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
 
 end OuterRegular
@@ -343,7 +343,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
         (inter_subset_right _ _).trans (disjointed_subset _ _),
         (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
     rw [← inter_Union, unionᵢ_disjointed, s.spanning, inter_univ]
-  rcases Ennreal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
+  rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
   rw [lt_tsub_iff_right, add_comm] at hδε
   have : ∀ n, ∃ (U : _)(_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n :=
     by
@@ -359,8 +359,8 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
   refine' ⟨⋃ n, U n, Union_mono hAU, isOpen_unionᵢ hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-    _ ≤ ∑' n, μ (A n) + δ n := Ennreal.tsum_le_tsum fun n => (hU n).le
-    _ = (∑' n, μ (A n)) + ∑' n, δ n := Ennreal.tsum_add
+    _ ≤ ∑' n, μ (A n) + δ n := ENNReal.tsum_le_tsum fun n => (hU n).le
+    _ = (∑' n, μ (A n)) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := congr_arg₂ (· + ·) (measure_Union hAd hAm).symm rfl
     _ < r := hδε
     
@@ -381,12 +381,12 @@ theorem measurableSetOfOpen [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0
   obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) :=
     by
     use (μ s - r) / 2
-    simp [*, hr.le, Ennreal.add_halves, Ennreal.sub_sub_cancel, le_add_right]
+    simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_is_open_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
   rcases(U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
   replace hsU' := diff_subset_comm.1 hsU'
   rcases H.exists_subset_lt_add h0 hUo hUt.ne hε with ⟨K, hKU, hKc, hKr⟩
-  refine' ⟨K \ U', fun x hx => hsU' ⟨hKU hx.1, hx.2⟩, hd hKc hU'o, Ennreal.sub_lt_of_lt_add hεs _⟩
+  refine' ⟨K \ U', fun x hx => hsU' ⟨hKU hx.1, hx.2⟩, hd hKc hU'o, ENNReal.sub_lt_of_lt_add hεs _⟩
   calc
     μ s ≤ μ U := μ.mono hsU
     _ < μ K + ε := hKr
@@ -442,8 +442,8 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
     simp only [measure_compl_le_add_iff, *, hUo.measurable_set, hFc.measurable_set, true_and_iff]
   -- check for disjoint unions
   · intro s hsd hsm H ε ε0
-    have ε0' : ε / 2 ≠ 0 := (Ennreal.half_pos ε0).ne'
-    rcases Ennreal.exists_pos_sum_of_countable' ε0' ℕ with ⟨δ, δ0, hδε⟩
+    have ε0' : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
+    rcases ENNReal.exists_pos_sum_of_countable' ε0' ℕ with ⟨δ, δ0, hδε⟩
     choose F hFs U hsU hFc hUo hF hU using fun n => H n (δ n) (δ0 n).ne'
     -- the approximating closed set is constructed by considering finitely many sets `s i`, which
     -- cover all the measure up to `ε/2`, approximating each of these by a closed set `F i`, and
@@ -452,7 +452,7 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
       by
       rw [measure_Union hsd hsm]
       exact tendsto.add ennreal.summable.has_sum tendsto_const_nhds
-    rcases(this.eventually <| lt_mem_nhds <| Ennreal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
+    rcases(this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
     -- the approximating open set is constructed by taking for each `s n` an approximating open set
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
     refine'
@@ -464,17 +464,17 @@ theorem weaklyRegularOfFinite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure
           rw [← sum_add_distrib]
           exact add_le_add_right (sum_le_sum fun k hk => hF k) _
         _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
-          add_le_add_right (add_le_add_left ((Ennreal.sum_le_tsum _).trans hδε.le) _) _
+          add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _
         _ = μ (⋃ k ∈ t, F k) + ε := _
         
-      rw [measure_bUnion_finset, add_assoc, Ennreal.add_halves]
+      rw [measure_bUnion_finset, add_assoc, ENNReal.add_halves]
       exacts[fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k hk => (hFc k).MeasurableSet]
     ·
       calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_Union_le _
-        _ ≤ ∑' n, μ (s n) + δ n := Ennreal.tsum_le_tsum hU
-        _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, Ennreal.tsum_add]
-        _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans Ennreal.half_le_self) _
+        _ ≤ ∑' n, μ (s n) + δ n := ENNReal.tsum_le_tsum hU
+        _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_Union hsd hsm, ENNReal.tsum_add]
+        _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
         
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegularOfFinite
 
@@ -530,7 +530,7 @@ theorem IsOpen.measure_eq_supᵢ_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ
 
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact,
-    Ennreal.supᵢ_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+    ENNReal.supᵢ_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a

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

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -431,7 +431,7 @@ lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set 
   refine' ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_iUnion_le _
-    _ ≤ ∑' n, (μ (A n) + δ n) := (ENNReal.tsum_le_tsum fun n => (hU n).le)
+    _ ≤ ∑' n, (μ (A n) + δ n) := ENNReal.tsum_le_tsum fun n => (hU n).le
     _ = ∑' n, μ (A n) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
     _ < r := hδε
@@ -481,7 +481,7 @@ theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOp
   calc
     μ s ≤ μ U := μ.mono hsU
     _ < μ K + ε := hKr
-    _ ≤ μ (K \ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)
+    _ ≤ μ (K \ U') + μ U' + ε := add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _
     _ ≤ μ (K \ U') + ε + ε := by
       apply add_le_add_right; apply add_le_add_left
       exact hμU'.le
@@ -549,7 +549,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
             fun k _ => (hFc k).measurableSet]
     · calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_iUnion_le _
-        _ ≤ ∑' n, (μ (s n) + δ n) := (ENNReal.tsum_le_tsum hU)
+        _ ≤ ∑' n, (μ (s n) + δ n) := ENNReal.tsum_le_tsum hU
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite

@@ -680,7 +680,7 @@ lemma innerRegularWRT_isClosed_isOpen [R1Space α] [OpensMeasurableSpace α] [h
     hK.trans_le (measure_mono subset_closure)⟩
 
 theorem exists_compact_not_null [InnerRegular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
-  simp_rw [Ne.def, ← measure_univ_eq_zero, MeasurableSet.univ.measure_eq_iSup_isCompact,
+  simp_rw [Ne, ← measure_univ_eq_zero, MeasurableSet.univ.measure_eq_iSup_isCompact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 
 /-- If `μ` is inner regular, then any measurable set can be approximated by a compact subset.
@@ -962,7 +962,7 @@ theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U)
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
-  simp_rw [Ne.def, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
+  simp_rw [Ne, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
 #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 

@@ -418,7 +418,7 @@ lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set 
     rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
   rw [lt_tsub_iff_right, add_comm] at hδε
-  have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
+  have : ∀ n, ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n := by
     intro n
     have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
     have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
@@ -470,7 +470,7 @@ theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOp
     have : 0 < μ univ := (bot_le.trans_lt hr).trans_le (measure_mono (subset_univ _))
     obtain ⟨K, -, hK, -⟩ : ∃ K, K ⊆ univ ∧ p K ∧ 0 < μ K := H isOpen_univ _ this
     simpa using hd hK isOpen_univ
-  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
+  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ ε ≠ 0, ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
     use (μ s - r) / 2
     simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right, tsub_eq_zero_iff_le]
   rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -472,7 +472,7 @@ theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOp
     simpa using hd hK isOpen_univ
   obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
     use (μ s - r) / 2
-    simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
+    simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right, tsub_eq_zero_iff_le]
   rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
   rcases (U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
   replace hsU' := diff_subset_comm.1 hsU'

• move IsCompact.exists_isOpen_lt_of_lt before IsCompact.measure_eq_iInf_isOpen;
• weaken TC assumptions from [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α] to [IsLocallyFiniteMeasure μ] [R1Space α].

Not sure if we have other ways to get a locally finite measure though.

@@ -775,43 +775,36 @@ instance (priority := 50) [BorelSpace α] [R1Space α] [h : InnerRegularCompactL
   innerRegular := InnerRegularWRT.trans h.innerRegular <|
     InnerRegularWRT.of_imp (fun U hU ↦ ⟨hU.measurableSet, measure_ne_top μ U⟩)
 
-/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
-compact space, then any compact set can be approximated from outside by open sets. -/
+protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
+    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α}
+    (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
+    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
+  rcases hK.exists_open_superset_measure_lt_top μ with ⟨V, hKV, hVo, hμV⟩
+  have := Fact.mk hμV
+  obtain ⟨U, hKU, hUo, hμU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict V U < r :=
+    exists_isOpen_lt_of_lt K r <| (restrict_apply_le _ _).trans_lt hr
+  refine ⟨U ∩ V, subset_inter hKU hKV, hUo.inter hVo, ?_⟩
+  rwa [restrict_apply hUo.measurableSet] at hμU
+
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets
+and is locally finite in an R₁ space,
+then any compact set can be approximated from outside by open sets. -/
 protected lemma _root_.IsCompact.measure_eq_iInf_isOpen [InnerRegularCompactLTTop μ]
-    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
-    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
+    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α} (hK : IsCompact K) :
     μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
   apply le_antisymm
   · simp only [le_iInf_iff]
-    rintro U KU -
-    exact measure_mono KU
-  apply le_of_forall_lt' (fun r hr ↦ ?_)
-  simp only [iInf_lt_iff, exists_prop, exists_and_left]
-  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
-    exists_compact_between hK isOpen_univ (subset_univ _)
-  have : Fact (μ (interior L) < ∞) :=
-    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
-  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
-    apply exists_isOpen_lt_of_lt K r
-    exact (restrict_apply_le _ _).trans_lt hr
-  refine ⟨U ∩ interior L, subset_inter KU KL, U_open.inter isOpen_interior, ?_⟩
-  rwa [restrict_apply U_open.measurableSet] at hU
+    exact fun U KU _ ↦ measure_mono KU
+  · apply le_of_forall_lt'
+    simpa only [iInf_lt_iff, exists_prop, exists_and_left] using hK.exists_isOpen_lt_of_lt
 
 @[deprecated] -- Since 28 Jan 2024
 alias _root_.IsCompact.measure_eq_infi_isOpen := IsCompact.measure_eq_iInf_isOpen
 
-protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
-    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
-    [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
-    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
-  have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
-    rwa [hK.measure_eq_iInf_isOpen] at hr
-  simpa only [iInf_lt_iff, exists_prop, exists_and_left]
-
 protected theorem _root_.IsCompact.exists_isOpen_lt_add [InnerRegularCompactLTTop μ]
-    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
-    [BorelSpace α] {K : Set α} (hK : IsCompact K) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-     ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < μ K + ε :=
+    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
+    {K : Set α} (hK : IsCompact K) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < μ K + ε :=
   hK.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hK.measure_lt_top.ne hε)
 
 instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by

• Remove Data.Set.Basic from scripts/noshake.json.
• Remove an exception that was used by examples only, move these examples to a new test file.
• Drop an exception for Order.Filter.Basic dependency on Control.Traversable.Instances, as the relevant parts were moved to Order.Filter.ListTraverse.
• Run lake exe shake --fix.

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
 -/
 import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
-import Mathlib.Topology.Metrizable.Urysohn
 
 #align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
 
@@ -1039,14 +1038,6 @@ instance (priority := 100) {X : Type*}
   have : Fact (μ (spanningSets μ n) < ∞) := ⟨measure_spanningSets_lt_top μ n⟩
   exact WeaklyRegular.innerRegular_measurable.trans InnerRegularWRT.of_sigmaFinite
 
-/- Check that typeclass inference works to guarantee regularity and inner regularity in
-interesting situations. -/
-example [LocallyCompactSpace α] [RegularSpace α] [BorelSpace α] [SecondCountableTopology α]
-    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] : Regular μ := by infer_instance
-
-example [LocallyCompactSpace α] [RegularSpace α] [BorelSpace α] [SecondCountableTopology α]
-    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] : InnerRegular μ := by infer_instance
-
 end Measure
 
 end MeasureTheory

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;

@@ -319,14 +319,13 @@ class InnerRegularCompactLTTop (μ : Measure α) : Prop where
   protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞)
 
 -- see Note [lower instance priority]
-/-- A regular measure is weakly regular in a T2 space or in a regular space. -/
-instance (priority := 100) Regular.weaklyRegular [T2OrLocallyCompactRegularSpace α] [Regular μ] :
-    WeaklyRegular μ := by
-  constructor
-  intro U hU r hr
-  rcases Regular.innerRegular hU r hr with ⟨K, KU, K_comp, hK⟩
-  exact ⟨closure K, K_comp.closure_subset_of_isOpen hU KU, isClosed_closure,
-    hK.trans_le (measure_mono subset_closure)⟩
+/-- A regular measure is weakly regular in an R₁ space. -/
+instance (priority := 100) Regular.weaklyRegular [R1Space α] [Regular μ] :
+    WeaklyRegular μ where
+  innerRegular := fun _U hU r hr ↦
+    let ⟨K, KU, K_comp, hK⟩ := Regular.innerRegular hU r hr
+    ⟨closure K, K_comp.closure_subset_of_isOpen hU KU, isClosed_closure,
+      hK.trans_le (measure_mono subset_closure)⟩
 #align measure_theory.measure.regular.weakly_regular MeasureTheory.Measure.Regular.weaklyRegular
 
 namespace OuterRegular
@@ -438,7 +437,9 @@ lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set 
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
     _ < r := hδε
 
-lemma measure_closure_eq_of_isCompact [T2OrLocallyCompactRegularSpace α] [OuterRegular μ]
+/-- See also `IsCompact.measure_closure` for a version
+that assumes the `σ`-algebra to be the Borel `σ`-algebra but makes no assumptions on `μ`. -/
+lemma measure_closure_eq_of_isCompact [R1Space α] [OuterRegular μ]
     {k : Set α} (hk : IsCompact k) : μ (closure k) = μ k := by
   apply le_antisymm ?_ (measure_mono subset_closure)
   simp only [measure_eq_iInf_isOpen k, le_iInf_iff]
@@ -672,8 +673,8 @@ instance smul_nnreal [InnerRegular μ] (c : ℝ≥0) : InnerRegular (c • μ) :
 instance (priority := 100) [InnerRegular μ] : InnerRegularCompactLTTop μ :=
   ⟨fun _s hs r hr ↦ InnerRegular.innerRegular hs.1 r hr⟩
 
-lemma innerRegularWRT_isClosed_isOpen [T2OrLocallyCompactRegularSpace α] [OpensMeasurableSpace α]
-    [h : InnerRegular μ] : InnerRegularWRT μ IsClosed IsOpen := by
+lemma innerRegularWRT_isClosed_isOpen [R1Space α] [OpensMeasurableSpace α] [h : InnerRegular μ] :
+    InnerRegularWRT μ IsClosed IsOpen := by
   intro U hU r hr
   rcases h.innerRegular hU.measurableSet r hr with ⟨K, KU, K_comp, hK⟩
   exact ⟨closure K, K_comp.closure_subset_of_isOpen hU KU, isClosed_closure,
@@ -766,20 +767,18 @@ instance (priority := 50) [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ]
   convert h.innerRegular with s
   simp [measure_ne_top μ s]
 
-instance (priority := 50) [BorelSpace α] [T2OrLocallyCompactRegularSpace α]
-    [InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : WeaklyRegular μ := by
-  apply InnerRegularWRT.weaklyRegular_of_finite
-  exact InnerRegular.innerRegularWRT_isClosed_isOpen
+instance (priority := 50) [BorelSpace α] [R1Space α] [InnerRegularCompactLTTop μ]
+    [IsFiniteMeasure μ] : WeaklyRegular μ :=
+  InnerRegular.innerRegularWRT_isClosed_isOpen.weaklyRegular_of_finite _
 
-instance (priority := 50) [BorelSpace α] [T2OrLocallyCompactRegularSpace α]
-    [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : Regular μ := by
-  constructor
-  apply InnerRegularWRT.trans h.innerRegular
-  exact InnerRegularWRT.of_imp (fun U hU ↦ ⟨hU.measurableSet, measure_ne_top μ U⟩)
+instance (priority := 50) [BorelSpace α] [R1Space α] [h : InnerRegularCompactLTTop μ]
+    [IsFiniteMeasure μ] : Regular μ where
+  innerRegular := InnerRegularWRT.trans h.innerRegular <|
+    InnerRegularWRT.of_imp (fun U hU ↦ ⟨hU.measurableSet, measure_ne_top μ U⟩)
 
 /-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
 compact space, then any compact set can be approximated from outside by open sets. -/
-protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
+protected lemma _root_.IsCompact.measure_eq_iInf_isOpen [InnerRegularCompactLTTop μ]
     [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
     [BorelSpace α] {K : Set α} (hK : IsCompact K) :
     μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
@@ -799,12 +798,15 @@ protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTo
   refine ⟨U ∩ interior L, subset_inter KU KL, U_open.inter isOpen_interior, ?_⟩
   rwa [restrict_apply U_open.measurableSet] at hU
 
+@[deprecated] -- Since 28 Jan 2024
+alias _root_.IsCompact.measure_eq_infi_isOpen := IsCompact.measure_eq_iInf_isOpen
+
 protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
     [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
     [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
     ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
   have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
-    rwa [hK.measure_eq_infi_isOpen] at hr
+    rwa [hK.measure_eq_iInf_isOpen] at hr
   simpa only [iInf_lt_iff, exists_prop, exists_and_left]
 
 protected theorem _root_.IsCompact.exists_isOpen_lt_add [InnerRegularCompactLTTop μ]
@@ -1006,7 +1008,7 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • 
 instance smul_nnreal [Regular μ] (c : ℝ≥0) : Regular (c • μ) := Regular.smul coe_ne_top
 
 /-- The restriction of a regular measure to a set of finite measure is regular. -/
-theorem restrict_of_measure_ne_top [T2OrLocallyCompactRegularSpace α] [BorelSpace α] [Regular μ]
+theorem restrict_of_measure_ne_top [R1Space α] [BorelSpace α] [Regular μ]
     {A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A) := by
   have : WeaklyRegular (μ.restrict A) := WeaklyRegular.restrict_of_measure_ne_top h'A
   constructor

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.

@@ -320,7 +320,7 @@ class InnerRegularCompactLTTop (μ : Measure α) : Prop where
 
 -- see Note [lower instance priority]
 /-- A regular measure is weakly regular in a T2 space or in a regular space. -/
-instance (priority := 100) Regular.weaklyRegular [ClosableCompactSubsetOpenSpace α] [Regular μ] :
+instance (priority := 100) Regular.weaklyRegular [T2OrLocallyCompactRegularSpace α] [Regular μ] :
     WeaklyRegular μ := by
   constructor
   intro U hU r hr
@@ -438,6 +438,13 @@ lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set 
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
     _ < r := hδε
 
+lemma measure_closure_eq_of_isCompact [T2OrLocallyCompactRegularSpace α] [OuterRegular μ]
+    {k : Set α} (hk : IsCompact k) : μ (closure k) = μ k := by
+  apply le_antisymm ?_ (measure_mono subset_closure)
+  simp only [measure_eq_iInf_isOpen k, le_iInf_iff]
+  intro u ku u_open
+  exact measure_mono (hk.closure_subset_of_isOpen u_open ku)
+
 end OuterRegular
 
 /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
@@ -665,7 +672,7 @@ instance smul_nnreal [InnerRegular μ] (c : ℝ≥0) : InnerRegular (c • μ) :
 instance (priority := 100) [InnerRegular μ] : InnerRegularCompactLTTop μ :=
   ⟨fun _s hs r hr ↦ InnerRegular.innerRegular hs.1 r hr⟩
 
-lemma innerRegularWRT_isClosed_isOpen [ClosableCompactSubsetOpenSpace α] [OpensMeasurableSpace α]
+lemma innerRegularWRT_isClosed_isOpen [T2OrLocallyCompactRegularSpace α] [OpensMeasurableSpace α]
     [h : InnerRegular μ] : InnerRegularWRT μ IsClosed IsOpen := by
   intro U hU r hr
   rcases h.innerRegular hU.measurableSet r hr with ⟨K, KU, K_comp, hK⟩
@@ -759,12 +766,12 @@ instance (priority := 50) [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ]
   convert h.innerRegular with s
   simp [measure_ne_top μ s]
 
-instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
+instance (priority := 50) [BorelSpace α] [T2OrLocallyCompactRegularSpace α]
     [InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : WeaklyRegular μ := by
   apply InnerRegularWRT.weaklyRegular_of_finite
   exact InnerRegular.innerRegularWRT_isClosed_isOpen
 
-instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
+instance (priority := 50) [BorelSpace α] [T2OrLocallyCompactRegularSpace α]
     [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : Regular μ := by
   constructor
   apply InnerRegularWRT.trans h.innerRegular
@@ -999,7 +1006,7 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • 
 instance smul_nnreal [Regular μ] (c : ℝ≥0) : Regular (c • μ) := Regular.smul coe_ne_top
 
 /-- The restriction of a regular measure to a set of finite measure is regular. -/
-theorem restrict_of_measure_ne_top [ClosableCompactSubsetOpenSpace α] [BorelSpace α] [Regular μ]
+theorem restrict_of_measure_ne_top [T2OrLocallyCompactRegularSpace α] [BorelSpace α] [Regular μ]
     {A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A) := by
   have : WeaklyRegular (μ.restrict A) := WeaklyRegular.restrict_of_measure_ne_top h'A
   constructor

@@ -423,7 +423,7 @@ lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set 
   have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
     intro n
     have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
-    have Ht : μ.restrict (s n) (A n) ≠ ⊤ := by
+    have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
       rw [H₁]
       exact ((measure_mono ((inter_subset_left _ _).trans (subset_iUnion A n))).trans_lt HA).ne
     rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
@@ -486,7 +486,7 @@ open Finset in
 sets. Then the measure is weakly regular. -/
 theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
     (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ := by
-  have hfin : ∀ {s}, μ s ≠ ⊤ := @(measure_ne_top μ)
+  have hfin : ∀ {s}, μ s ≠ ∞ := @(measure_ne_top μ)
   suffices ∀ s, MeasurableSet s → ∀ ε, ε ≠ 0 → ∃ F, F ⊆ s ∧ ∃ U, U ⊇ s ∧
       IsClosed F ∧ IsOpen U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε by
     refine'

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.

@@ -221,7 +221,7 @@ theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
 
 theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ K, K ⊆ U ∧ p K ∧ μ U < μ K + ε := by
-  cases' eq_or_ne (μ U) 0 with h₀ h₀
+  rcases eq_or_ne (μ U) 0 with h₀ | h₀
   · refine' ⟨∅, empty_subset _, h0, _⟩
     rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
   · rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -60,7 +60,7 @@ satisfying a predicate `q` with respect to sets satisfying a predicate `p` if fo
 `U ∈ {U | q U}` and a number `r < μ U` there exists `F ⊆ U` such that `p F` and `r < μ F`.
 
 There are two main nontrivial results in the development below:
-* `InnerRegularWRT.measurableSet_of_open` shows that, for an outer regular measure, inner
+* `InnerRegularWRT.measurableSet_of_isOpen` shows that, for an outer regular measure, inner
 regularity for open sets with respect to compact sets or closed sets implies inner regularity for
 all measurable sets of finite measure (with respect to compact sets or closed sets respectively).
 * `InnerRegularWRT.weaklyRegular_of_finite` shows that a finite measure which is inner regular for
@@ -455,7 +455,7 @@ variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
 /-- If a measure is inner regular (using closed or compact sets) for open sets, then every
 measurable set of finite measure can be approximated by a (closed or compact) subset. -/
-theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen)
+theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
     InnerRegularWRT μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ := by
   rintro s ⟨hs, hμs⟩ r hr
@@ -479,7 +479,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen
       apply add_le_add_right; apply add_le_add_left
       exact hμU'.le
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
-#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open
+#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen
 
 open Finset in
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
@@ -863,7 +863,7 @@ theorem _root_.IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U)
 
 theorem innerRegular_measurable [WeaklyRegular μ] :
     InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  WeaklyRegular.innerRegular.measurableSet_of_open (fun _ _ h₁ h₂ ↦ h₁.inter h₂.isClosed_compl)
+  WeaklyRegular.innerRegular.measurableSet_of_isOpen (fun _ _ h₁ h₂ ↦ h₁.inter h₂.isClosed_compl)
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
@@ -969,7 +969,7 @@ theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠
 compact subset. See also `MeasurableSet.exists_isCompact_lt_add` and
 `MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
 instance (priority := 100) [Regular μ] : InnerRegularCompactLTTop μ :=
-  ⟨Regular.innerRegular.measurableSet_of_open (fun _ _ hs hU ↦ hs.diff hU)⟩
+  ⟨Regular.innerRegular.measurableSet_of_isOpen (fun _ _ hs hU ↦ hs.diff hU)⟩
 #noalign measure_theory.measure.regular.inner_regular_measurable
 
 protected theorem map [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]

Towards general uniqueness results for the Haar measure, we introduce a new class of regular measures called InnerRegular, for measures which are inner regular with respect to compact sets. We also introduce InnerRegularWRT for more general classes of inner regular measures with properties to be prescribed, and InnerRegularCompactLTTop for measures which are regular for finite measure sets with respect to compact sets -- the latter property is the common denominator to the two main classes of Haar measures, the regular ones and the inner regular ones.

@@ -1,9 +1,10 @@
 /-
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Floris Van Doorn, Yury Kudryashov
+Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
 -/
 import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
+import Mathlib.Topology.Metrizable.Urysohn
 
 #align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
 
@@ -13,42 +14,77 @@ import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
 A measure is `OuterRegular` if the measure of any measurable set `A` is the infimum of `μ U` over
 all open sets `U` containing `A`.
 
-A measure is `Regular` if it satisfies the following properties:
-* it is finite on compact sets;
-* it is outer regular;
-* it is inner regular for open sets with respect to compacts sets: the measure of any open set `U`
-  is the supremum of `μ K` over all compact sets `K` contained in `U`.
-
 A measure is `WeaklyRegular` if it satisfies the following properties:
 * it is outer regular;
 * it is inner regular for open sets with respect to closed sets: the measure of any open set `U`
   is the supremum of `μ F` over all closed sets `F` contained in `U`.
 
-In a Hausdorff topological space, regularity implies weak regularity. These three conditions are
-registered as typeclasses for a measure `μ`, and this implication is recorded as an instance.
-
-In order to avoid code duplication, we also define a measure `μ` to be `InnerRegular` for sets
+A measure is `Regular` if it satisfies the following properties:
+* it is finite on compact sets;
+* it is outer regular;
+* it is inner regular for open sets with respect to compacts closed sets: the measure of any open
+  set `U` is the supremum of `μ K` over all compact sets `K` contained in `U`.
+
+A measure is `InnerRegular` if it is inner regular for measurable sets with respect to compact
+sets: the measure of any measurable set `s` is the supremum of `μ K` over all compact
+sets contained in `s`.
+
+A measure is `InnerRegularCompactLTTop` if it is inner regular for measurable sets of finite
+measure with respect to compact sets: the measure of any measurable set `s` is the supremum
+of `μ K` over all compact sets contained in `s`.
+
+There is a reason for this zoo of regularity classes:
+* A finite measure on a metric space is always weakly regular. Therefore, in probability theory,
+  weakly regular measures play a prominent role.
+* In locally compact topological spaces, there are two competing notions of Radon measures: the
+  ones that are regular, and the ones that are inner regular. For any of these two notions, there is
+  a Riesz representation theorem, and an existence and uniqueness statement for the Haar measure in
+  locally compact topological groups. The two notions coincide in sigma-compact spaces, but they
+  differ in general, so it is worth having the two of them.
+* Both notions of Haar measure satisfy the weaker notion `InnerRegularCompactLTTop`, so it is worth
+  trying to express theorems using this weaker notion whenever possible, to make sure that it
+  applies to both Haar measures simultaneously.
+
+While traditional textbooks on measure theory on locally compact spaces emphasize regular measures,
+more recent textbooks emphasize that inner regular Haar measures are better behaved than regular
+Haar measures, so we will develop both notions.
+
+The five conditions above are registered as typeclasses for a measure `μ`, and implications between
+them are recorded as instances. For example, in a Hausdorff topological space, regularity implies
+weak regularity. Also, regularity or inner regularity both imply `InnerRegularCompactLTTop`.
+In a regular locally compact finite measure space, then regularity, inner regularity
+and `InnerRegularCompactLTTop` are all equivalent.
+
+In order to avoid code duplication, we also define a measure `μ` to be `InnerRegularWRT` for sets
 satisfying a predicate `q` with respect to sets satisfying a predicate `p` if for any set
 `U ∈ {U | q U}` and a number `r < μ U` there exists `F ⊆ U` such that `p F` and `r < μ F`.
 
-We prove that inner regularity for open sets with respect to compact sets or closed sets implies
-inner regularity for all measurable sets of finite measure (with respect to
-compact sets or closed sets respectively), and register some corollaries for (weakly) regular
-measures.
-
-Note that a similar statement for measurable sets of infinite mass can fail. For a counterexample,
-consider the group `ℝ × ℝ` where the first factor has the discrete topology and the second one the
-usual topology. It is a locally compact Hausdorff topological group, with Haar measure equal to
-Lebesgue measure on each vertical fiber. The set `ℝ × {0}` has infinite measure (by outer
-regularity), but any compact set it contains has zero measure (as it is finite).
-
-Several authors require as a definition of regularity that all measurable sets are inner regular.
-We have opted for the slightly weaker definition above as it holds for all Haar measures, it is
-enough for essentially all applications, and it is equivalent to the other definition when the
-measure is finite.
-
-The interest of the notion of weak regularity is that it is enough for many applications, and it
-is automatically satisfied by any finite measure on a metric space.
+There are two main nontrivial results in the development below:
+* `InnerRegularWRT.measurableSet_of_open` shows that, for an outer regular measure, inner
+regularity for open sets with respect to compact sets or closed sets implies inner regularity for
+all measurable sets of finite measure (with respect to compact sets or closed sets respectively).
+* `InnerRegularWRT.weaklyRegular_of_finite` shows that a finite measure which is inner regular for
+open sets with respect to closed sets (for instance a finite measure on a metric space) is weakly
+regular.
+
+All other results are deduced from these ones.
+
+Here is an example showing how regularity and inner regularity may differ even on locally compact
+spaces. Consider the group `ℝ × ℝ` where the first factor has the discrete topology and the second
+one the usual topology. It is a locally compact Hausdorff topological group, with Haar measure equal
+to Lebesgue measure on each vertical fiber. Let us consider the regular version of Haar measure.
+Then the set `ℝ × {0}` has infinite measure (by outer regularity), but any compact set it contains
+has zero measure (as it is finite). In fact, this set only contains subset with measure zero or
+infinity. The inner regular version of Haar measure, on the other hand, gives zero mass to the
+set `ℝ × {0}`.
+
+Another interesting example is the sum of the Dirac masses at rational points in the real line.
+It is a σ-finite measure on a locally compact metric space, but it is not outer regular: for
+outer regularity, one needs additional locally finite assumptions. On the other hand, it is
+inner regular.
+
+Several authors require both regularity and inner regularity for their measures. We have opted
+for the more fine grained definitions above as they apply more generally.
 
 ## Main definitions
 
@@ -58,8 +94,13 @@ is automatically satisfied by any finite measure on a metric space.
   space is regular.
 * `MeasureTheory.Measure.WeaklyRegular μ`: a typeclass registering that a measure `μ` on a
   topological space is weakly regular.
-* `MeasureTheory.Measure.InnerRegular μ p q`: a non-typeclass predicate saying that a measure `μ`
+* `MeasureTheory.Measure.InnerRegularWRT μ p q`: a non-typeclass predicate saying that a measure `μ`
   is inner regular for sets satisfying `q` with respect to sets satisfying `p`.
+* `MeasureTheory.Measure.InnerRegular μ`: a typeclass registering that a measure `μ` on a
+  topological space is inner regular for measurable sets with respect to compact sets.
+* `MeasureTheory.Measure.InnerRegularCompactLTTop μ`: a typeclass registering that a measure `μ`
+  on a topological space is inner regular for measurable sets of finite measure with respect to
+  compact sets.
 
 ## Main results
 
@@ -83,12 +124,12 @@ is automatically satisfied by any finite measure on a metric space.
 *  `MeasurableSet.exists_lt_isClosed_of_ne_top` and `MeasurableSet.exists_isClosed_lt_add`:
   a measurable set of finite measure can be approximated by a closed subset (stated as
   `r < μ F` and `μ s < μ F + ε`, respectively).
-* `MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure` is an
+* `MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_of_isFiniteMeasure` is an
   instance registering that a finite measure on a metric space is weakly regular (in fact, a pseudo
-  emetric space is enough);
-* `MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite`
+  metrizable space is enough);
+* `MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_secondCountable_of_locallyFinite`
   is an instance registering that a locally finite measure on a second countable metric space (or
-  even a pseudo emetric space) is weakly regular.
+  even a pseudo metrizable space) is weakly regular.
 
 ### Regular measures
 
@@ -96,14 +137,24 @@ is automatically satisfied by any finite measure on a metric space.
   the measure of compact sets it contains.
 * `IsOpen.exists_lt_isCompact`: for an open set `U` and `r < μ U`, there exists a compact `K ⊆ U`
   of measure greater than `r`;
+* `MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure` is an
+  instance registering that a locally finite measure on a `σ`-compact metric space is regular (in
+  fact, an emetric space is enough).
+
+### Inner regular measures
+
+* `MeasurableSet.measure_eq_iSup_isCompact` asserts that the measure of a measurable set is the
+  supremum of the measure of compact sets it contains.
+* `MeasurableSet.exists_lt_isCompact`: for a measurable set `s` and `r < μ s`, there exists a
+  compact `K ⊆ s` of measure greater than `r`;
+
+### Inner regular measures for finite measure sets with respect to compact sets
+
 * `MeasurableSet.measure_eq_iSup_isCompact_of_ne_top` asserts that the measure of a measurable set
   of finite measure is the supremum of the measure of compact sets it contains.
 *  `MeasurableSet.exists_lt_isCompact_of_ne_top` and `MeasurableSet.exists_isCompact_lt_add`:
   a measurable set of finite measure can be approximated by a compact subset (stated as
   `r < μ K` and `μ s < μ K + ε`, respectively).
-* `MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure` is an
-  instance registering that a locally finite measure on a `σ`-compact metric space is regular (in
-  fact, an emetric space is enough).
 
 ## Implementation notes
 
@@ -119,6 +170,16 @@ Once this statement is proved, one deduces results for `σ`-finite measures from
 restricting them to finite measure sets (and proving that this restriction is weakly regular, using
 again the same statement).
 
+For non-Hausdorff spaces, one may argue whether the right condition for inner regularity is with
+respect to compact sets, or to compact closed sets. For instance,
+[Fremlin, *Measure Theory* (volume 4, 411J)][fremlin_vol4] considers measures which are inner
+regular with respect to compact closed sets (and calls them *tight*). However, since most of the
+literature uses mere compact sets, we have chosen to follow this convention. It doesn't make a
+difference in Hausdorff spaces, of course. In locally compact topological groups, the two
+conditions coincide, since if a compact set `k` is contained in a measurable set `u`, then the
+closure of `k` is a compact closed set still contained in `u`, see
+`IsCompact.closure_subset_of_measurableSet_of_group`.
+
 ## References
 
 [Halmos, Measure Theory, §52][halmos1950measure]. Note that Halmos uses an unusual definition of
@@ -126,10 +187,11 @@ Borel sets (for him, they are elements of the `σ`-algebra generated by compact
 proofs or statements do not apply directly.
 
 [Billingsley, Convergence of Probability Measures][billingsley1999]
--/
 
+[Bogachev, Measure Theory, volume 2, Theorem 7.11.1][bogachev2007]
+-/
 
-open Set Filter ENNReal Topology NNReal BigOperators
+open Set Filter ENNReal Topology NNReal BigOperators TopologicalSpace
 
 namespace MeasureTheory
 
@@ -141,57 +203,78 @@ of measure greater than `r`.
 
 This definition is used to prove some facts about regular and weakly regular measures without
 repeating the proofs. -/
-def InnerRegular {α} {_ : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
+def InnerRegularWRT {α} {_ : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) :=
   ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ K, K ⊆ U ∧ p K ∧ r < μ K
-#align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegular
+#align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegularWRT
 
-namespace InnerRegular
+namespace InnerRegularWRT
 
 variable {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
-theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
+theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
     μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K := by
   refine'
     le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
-#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup
+#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup
 
-theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
+theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
     (hε : ε ≠ 0) : ∃ K, K ⊆ U ∧ p K ∧ μ U < μ K + ε := by
   cases' eq_or_ne (μ U) 0 with h₀ h₀
   · refine' ⟨∅, empty_subset _, h0, _⟩
     rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
   · rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
     exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
-#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add
+#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add
 
-theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
-    (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
+protected theorem map {α β} [MeasurableSpace α] [MeasurableSpace β]
+    {μ : Measure α} {pa qa : Set α → Prop}
+    (H : InnerRegularWRT μ pa qa) {f : α → β} (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
     (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
-    (hB₁ : ∀ K, pb K → MeasurableSet K) (hB₂ : ∀ U, qb U → MeasurableSet U) :
-    InnerRegular (map f μ) pb qb := by
+    (hB₂ : ∀ U, qb U → MeasurableSet U) :
+    InnerRegularWRT (map f μ) pb qb := by
   intro U hU r hr
   rw [map_apply_of_aemeasurable hf (hB₂ _ hU)] at hr
   rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
+  refine ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, ?_⟩
+  exact hK.trans_le (le_map_apply_image hf _)
+#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegularWRT.map
+
+theorem map' {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
+    (H : InnerRegularWRT μ pa qa) (f : α ≃ᵐ β) {pb qb : Set β → Prop}
+    (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K)) :
+    InnerRegularWRT (map f μ) pb qb := by
+  intro U hU r hr
+  rw [f.map_apply U] at hr
+  rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
   refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
-  rwa [map_apply_of_aemeasurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
-#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map
+  rwa [f.map_apply, f.preimage_image]
 
-theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q := by
+theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q := by
   intro U hU r hr
   rw [smul_apply, H.measure_eq_iSup hU, smul_eq_mul] at hr
   simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
-#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul
+#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegularWRT.smul
 
-theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') :
-    InnerRegular μ p q' := by
+theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
+    InnerRegularWRT μ p q' := by
   intro U hU r hr
   rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
   exact ⟨K, hKF.trans hFU, hpK, hrK⟩
-#align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegular.trans
+#align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegularWRT.trans
 
-end InnerRegular
+theorem rfl {p : Set α → Prop} : InnerRegularWRT μ p p :=
+  fun U hU _r hr ↦ ⟨U, Subset.rfl, hU, hr⟩
+
+theorem of_imp (h : ∀ s, q s → p s) : InnerRegularWRT μ p q :=
+  fun U hU _ hr ↦ ⟨U, Subset.rfl, h U hU, hr⟩
+
+theorem mono {p' q' : Set α → Prop} (H : InnerRegularWRT μ p q)
+    (h : ∀ s, q' s → q s) (h' : ∀ s, p s → p' s) : InnerRegularWRT μ p' q' :=
+  of_imp h' |>.trans H |>.trans (of_imp h)
+
+end InnerRegularWRT
 
 variable {α β : Type*} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
@@ -211,7 +294,7 @@ class OuterRegular (μ : Measure α) : Prop where
   - it is inner regular for open sets, using compact sets:
     `μ(U) = sup {μ(K) | K ⊆ U compact}` for `U` open. -/
 class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
-  innerRegular : InnerRegular μ IsCompact IsOpen
+  innerRegular : InnerRegularWRT μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
 
 /-- A measure `μ` is weakly regular if
@@ -219,16 +302,31 @@ class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegul
   - it is inner regular for open sets, using closed sets:
     `μ(U) = sup {μ(F) | F ⊆ U closed}` for `U` open. -/
 class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
-  protected innerRegular : InnerRegular μ IsClosed IsOpen
+  protected innerRegular : InnerRegularWRT μ IsClosed IsOpen
 #align measure_theory.measure.weakly_regular MeasureTheory.Measure.WeaklyRegular
 #align measure_theory.measure.weakly_regular.inner_regular MeasureTheory.Measure.WeaklyRegular.innerRegular
 
+/-- A measure `μ` is inner regular if, for any measurable set `s`, then
+`μ(s) = sup {μ(K) | K ⊆ s compact}`. -/
+class InnerRegular (μ : Measure α) : Prop where
+  protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s)
+
+/-- A measure `μ` is inner regular for finite measure sets with respect to compact sets:
+for any measurable set `s` with finite measure, then `μ(s) = sup {μ(K) | K ⊆ s compact}`.
+The main interest of this class is that it is satisfied for both natural Haar measures (the
+regular one and the inner regular one). -/
+class InnerRegularCompactLTTop (μ : Measure α) : Prop where
+  protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞)
+
 -- see Note [lower instance priority]
-/-- A regular measure is weakly regular. -/
-instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : WeaklyRegular μ where
-  innerRegular _U hU r hr :=
-    let ⟨K, hKU, hcK, hK⟩ := Regular.innerRegular hU r hr
-    ⟨K, hKU, hcK.isClosed, hK⟩
+/-- A regular measure is weakly regular in a T2 space or in a regular space. -/
+instance (priority := 100) Regular.weaklyRegular [ClosableCompactSubsetOpenSpace α] [Regular μ] :
+    WeaklyRegular μ := by
+  constructor
+  intro U hU r hr
+  rcases Regular.innerRegular hU r hr with ⟨K, KU, K_comp, hK⟩
+  exact ⟨closure K, K_comp.closure_subset_of_isOpen hU KU, isClosed_closure,
+    hK.trans_le (measure_mono subset_closure)⟩
 #align measure_theory.measure.regular.weakly_regular MeasureTheory.Measure.Regular.weaklyRegular
 
 namespace OuterRegular
@@ -298,37 +396,39 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx
     simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr
 #align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smul
 
-end OuterRegular
+instance smul_nnreal (μ : Measure α) [OuterRegular μ] (c : ℝ≥0) :
+    OuterRegular (c • μ) :=
+  OuterRegular.smul μ coe_ne_top
 
-/-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
-is outer regular, then the original measure is outer regular as well. -/
-protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
-    (s : μ.FiniteSpanningSetsIn { U | IsOpen U ∧ OuterRegular (μ.restrict U) }) :
+/-- If the restrictions of a measure to countably many open sets covering the space are
+outer regular, then the measure itself is outer regular. -/
+lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
+    (h : ∀ n, OuterRegular (μ.restrict (s n))) (h' : ∀ n, IsOpen (s n)) (h'' : univ ⊆ ⋃ n, s n) :
     OuterRegular μ := by
   refine' ⟨fun A hA r hr => _⟩
-  have hm : ∀ n, MeasurableSet (s.set n) := fun n => (s.set_mem n).1.measurableSet
-  haveI : ∀ n, OuterRegular (μ.restrict (s.set n)) := fun n => (s.set_mem n).2
+  have HA : μ A < ∞ := lt_of_lt_of_le hr le_top
+  have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
   -- Note that `A = ⋃ n, A ∩ disjointed s n`. We replace `A` with this sequence.
   obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
     ∃ A' : ℕ → Set α,
       (∀ n, MeasurableSet (A' n)) ∧
-        (∀ n, A' n ⊆ s.set n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
+        (∀ n, A' n ⊆ s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
     refine'
-      ⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
+      ⟨fun n => A ∩ disjointed s n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
         (inter_subset_right _ _).trans (disjointed_subset _ _),
-        (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
-    rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]
+        (disjoint_disjointed s).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
+    rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
   rw [lt_tsub_iff_right, add_comm] at hδε
   have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
     intro n
-    have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
-    have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
+    have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
+    have Ht : μ.restrict (s n) (A n) ≠ ⊤ := by
       rw [H₁]
-      exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne
+      exact ((measure_mono ((inter_subset_left _ _).trans (subset_iUnion A n))).trans_lt HA).ne
     rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
     rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
-    exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
+    exact ⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩
   choose U hAU hUo hU using this
   refine' ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _⟩
   calc
@@ -337,18 +437,32 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     _ = ∑' n, μ (A n) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
     _ < r := hδε
+
+end OuterRegular
+
+/-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set
+is outer regular, then the original measure is outer regular as well. -/
+protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
+    (s : μ.FiniteSpanningSetsIn { U | IsOpen U ∧ OuterRegular (μ.restrict U) }) :
+    OuterRegular μ :=
+  OuterRegular.of_restrict (s := fun n ↦ s.set n) (fun n ↦ (s.set_mem n).2)
+    (fun n ↦ (s.set_mem n).1) s.spanning.symm.subset
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
 
-namespace InnerRegular
+namespace InnerRegularWRT
 
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
-/-- If a measure is inner regular (using closed or compact sets), then every measurable set of
-finite measure can be approximated by a (closed or compact) subset. -/
-theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
+/-- If a measure is inner regular (using closed or compact sets) for open sets, then every
+measurable set of finite measure can be approximated by a (closed or compact) subset. -/
+theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
-    InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ := by
+    InnerRegularWRT μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ := by
   rintro s ⟨hs, hμs⟩ r hr
+  have h0 : p ∅ := by
+    have : 0 < μ univ := (bot_le.trans_lt hr).trans_le (measure_mono (subset_univ _))
+    obtain ⟨K, -, hK, -⟩ : ∃ K, K ⊆ univ ∧ p K ∧ 0 < μ K := H isOpen_univ _ this
+    simpa using hd hK isOpen_univ
   obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
     use (μ s - r) / 2
     simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
@@ -365,14 +479,13 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
       apply add_le_add_right; apply add_le_add_left
       exact hμU'.le
     _ = μ (K \ U') + (ε + ε) := add_assoc _ _ _
-#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_open
-
-open Finset
+#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_open
 
+open Finset in
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
 theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
-    (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ := by
+    (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ := by
   have hfin : ∀ {s}, μ s ≠ ⊤ := @(measure_ne_top μ)
   suffices ∀ s, MeasurableSet s → ∀ ε, ε ≠ 0 → ∃ F, F ⊆ s ∧ ∃ U, U ⊇ s ∧
       IsClosed F ∧ IsOpen U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε by
@@ -432,22 +545,40 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
         _ ≤ ∑' n, (μ (s n) + δ n) := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
-#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite
+#align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite
+
+/-- If the restrictions of a measure to a monotone sequence of sets covering the space are
+inner regular for some property `p` and all measurable sets, then the measure itself is
+inner regular. -/
+lemma of_restrict {μ : Measure α} {s : ℕ → Set α}
+    (h : ∀ n, InnerRegularWRT (μ.restrict (s n)) p MeasurableSet)
+    (hs : univ ⊆ ⋃ n, s n) (hmono : Monotone s) : InnerRegularWRT μ p MeasurableSet := by
+  intro F hF r hr
+  have hBU : ⋃ n, F ∩ s n = F := by  rw [← inter_iUnion, univ_subset_iff.mp hs, inter_univ]
+  have : μ F = ⨆ n, μ (F ∩ s n) := by
+    rw [← measure_iUnion_eq_iSup, hBU]
+    exact Monotone.directed_le fun m n h ↦ inter_subset_inter_right _ (hmono h)
+  rw [this] at hr
+  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
+  rw [← restrict_apply hF] at hn
+  rcases h n hF _ hn with ⟨K, KF, hKp, hK⟩
+  exact ⟨K, KF, hKp, hK.trans_le (restrict_apply_le _ _)⟩
 
-/-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
-by closed sets. -/
-theorem of_pseudoEMetricSpace {X : Type*} [PseudoEMetricSpace X] [MeasurableSpace X]
-    (μ : Measure X) : InnerRegular μ IsClosed IsOpen := by
+/-- In a metrizable space (or even a pseudo metrizable space), an open set can be approximated from
+inside by closed sets. -/
+theorem of_pseudoMetrizableSpace {X : Type*} [TopologicalSpace X] [PseudoMetrizableSpace X]
+    [MeasurableSpace X] (μ : Measure X) : InnerRegularWRT μ IsClosed IsOpen := by
+  let A : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
   intro U hU r hr
   rcases hU.exists_iUnion_isClosed with ⟨F, F_closed, -, rfl, F_mono⟩
   rw [measure_iUnion_eq_iSup F_mono.directed_le] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_iUnion _ _, F_closed n, hn⟩
-#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
+#align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace
 
 /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/
 theorem isCompact_isClosed {X : Type*} [TopologicalSpace X] [SigmaCompactSpace X]
-    [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed := by
+    [MeasurableSpace X] (μ : Measure X) : InnerRegularWRT μ IsCompact IsClosed := by
   intro F hF r hr
   set B : ℕ → Set X := compactCovering X
   have hBc : ∀ n, IsCompact (F ∩ B n) := fun n => (isCompact_compactCovering X n).inter_left hF
@@ -458,53 +589,139 @@ theorem isCompact_isClosed {X : Type*} [TopologicalSpace X] [SigmaCompactSpace X
   rw [this] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left _ _, hBc n, hn⟩
-#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed
+#align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed
+
+/-- If `μ` is inner regular for measurable finite measure sets with respect to some class of sets,
+then its restriction to any set is also inner regular for measurable finite measure sets, with
+respect to the same class of sets. -/
+lemma restrict (h : InnerRegularWRT μ p (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞)) (A : Set α) :
+    InnerRegularWRT (μ.restrict A) p (fun s ↦ MeasurableSet s ∧ μ.restrict A s ≠ ∞) := by
+  rintro s ⟨s_meas, hs⟩ r hr
+  rw [restrict_apply s_meas] at hs
+  obtain ⟨K, K_subs, pK, rK⟩ : ∃ K, K ⊆ (toMeasurable μ (s ∩ A)) ∩ s ∧ p K ∧ r < μ K := by
+    have : r < μ ((toMeasurable μ (s ∩ A)) ∩ s) := by
+      apply hr.trans_le
+      rw [restrict_apply s_meas]
+      exact measure_mono <| subset_inter (subset_toMeasurable μ (s ∩ A)) (inter_subset_left _ _)
+    refine h ⟨(measurableSet_toMeasurable _ _).inter s_meas, ?_⟩ _ this
+    apply (lt_of_le_of_lt _ hs.lt_top).ne
+    rw [← measure_toMeasurable (s ∩ A)]
+    exact measure_mono (inter_subset_left _ _)
+  refine ⟨K, K_subs.trans (inter_subset_right _ _), pK, ?_⟩
+  calc
+  r < μ K := rK
+  _ = μ.restrict (toMeasurable μ (s ∩ A)) K := by
+    rw [restrict_apply' (measurableSet_toMeasurable μ (s ∩ A))]
+    congr
+    apply (inter_eq_left.2 ?_).symm
+    exact K_subs.trans (inter_subset_left _ _)
+  _ = μ.restrict (s ∩ A) K := by rwa [restrict_toMeasurable]
+  _ ≤ μ.restrict A K := Measure.le_iff'.1 (restrict_mono (inter_subset_right _ _) le_rfl) K
+
+/-- If `μ` is inner regular for measurable finite measure sets with respect to some class of sets,
+then its restriction to any finite measure set is also inner regular for measurable sets with
+respect to the same class of sets. -/
+lemma restrict_of_measure_ne_top (h : InnerRegularWRT μ p (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞))
+    {A : Set α} (hA : μ A ≠ ∞) :
+    InnerRegularWRT (μ.restrict A) p (fun s ↦ MeasurableSet s) := by
+  have : Fact (μ A < ∞) := ⟨hA.lt_top⟩
+  exact (restrict h A).trans (of_imp (fun s hs ↦ ⟨hs, measure_ne_top _ _⟩))
+
+/-- Given a σ-finite measure, any measurable set can be approximated from inside by a measurable
+set of finite measure. -/
+lemma of_sigmaFinite [SigmaFinite μ] :
+    InnerRegularWRT μ (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) (fun s ↦ MeasurableSet s) := by
+  intro s hs r hr
+  set B : ℕ → Set α := spanningSets μ
+  have hBU : ⋃ n, s ∩ B n = s := by rw [← inter_iUnion, iUnion_spanningSets, inter_univ]
+  have : μ s = ⨆ n, μ (s ∩ B n) := by
+    rw [← measure_iUnion_eq_iSup, hBU]
+    exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (monotone_spanningSets μ h)
+  rw [this] at hr
+  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
+  refine ⟨s ∩ B n, inter_subset_left _ _, ⟨hs.inter (measurable_spanningSets μ n), ?_⟩, hn⟩
+  exact ((measure_mono (inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top μ n)).ne
 
-end InnerRegular
+end InnerRegularWRT
 
-namespace Regular
+namespace InnerRegular
 
-instance zero : Regular (0 : Measure α) :=
+variable {U : Set α} {ε : ℝ≥0∞}
+
+/-- The measure of a measurable set is the supremum of the measures of compact sets it contains. -/
+theorem _root_.MeasurableSet.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : MeasurableSet U)
+    (μ : Measure α) [InnerRegular μ] :
+    μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K :=
+  InnerRegular.innerRegular.measure_eq_iSup hU
+
+instance zero : InnerRegular (0 : Measure α) :=
   ⟨fun _ _ _r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩
-#align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
 
-/-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
-theorem _root_.IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
-    (hr : r < μ U) : ∃ K, K ⊆ U ∧ IsCompact K ∧ r < μ K :=
-  Regular.innerRegular hU r hr
-#align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
+instance smul [h : InnerRegular μ] (c : ℝ≥0∞) : InnerRegular (c • μ) :=
+  ⟨InnerRegularWRT.smul h.innerRegular c⟩
 
-/-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
-theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
-    [Regular μ] : μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K :=
-  Regular.innerRegular.measure_eq_iSup hU
-#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
+instance smul_nnreal [InnerRegular μ] (c : ℝ≥0) : InnerRegular (c • μ) := smul (c : ℝ≥0∞)
 
-theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
-  simp_rw [Ne.def, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
+instance (priority := 100) [InnerRegular μ] : InnerRegularCompactLTTop μ :=
+  ⟨fun _s hs r hr ↦ InnerRegular.innerRegular hs.1 r hr⟩
+
+lemma innerRegularWRT_isClosed_isOpen [ClosableCompactSubsetOpenSpace α] [OpensMeasurableSpace α]
+    [h : InnerRegular μ] : InnerRegularWRT μ IsClosed IsOpen := by
+  intro U hU r hr
+  rcases h.innerRegular hU.measurableSet r hr with ⟨K, KU, K_comp, hK⟩
+  exact ⟨closure K, K_comp.closure_subset_of_isOpen hU KU, isClosed_closure,
+    hK.trans_le (measure_mono subset_closure)⟩
+
+theorem exists_compact_not_null [InnerRegular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
+  simp_rw [Ne.def, ← measure_univ_eq_zero, MeasurableSet.univ.measure_eq_iSup_isCompact,
     ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
-#align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
 
-/-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
-compact subset. See also `MeasurableSet.exists_isCompact_lt_add` and
-`MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
-theorem innerRegular_measurable [Regular μ] :
-    InnerRegular μ IsCompact fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff
-#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable
+/-- If `μ` is inner regular, then any measurable set can be approximated by a compact subset.
+See also `MeasurableSet.exists_isCompact_lt_add_of_ne_top`. -/
+theorem _root_.MeasurableSet.exists_lt_isCompact [InnerRegular μ] ⦃A : Set α⦄
+    (hA : MeasurableSet A) {r : ℝ≥0∞} (hr : r < μ A) :
+    ∃ K, K ⊆ A ∧ IsCompact K ∧ r < μ K :=
+  InnerRegular.innerRegular hA _ hr
+
+protected theorem map_of_continuous [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] [h : InnerRegular μ] {f : α → β} (hf : Continuous f) :
+    InnerRegular (Measure.map f μ) :=
+  ⟨InnerRegularWRT.map h.innerRegular hf.aemeasurable (fun _s hs ↦ hf.measurable hs)
+    (fun _K hK ↦ hK.image hf) (fun _s hs ↦ hs)⟩
+
+protected theorem map [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] [InnerRegular μ] (f : α ≃ₜ β) : (Measure.map f μ).InnerRegular :=
+  InnerRegular.map_of_continuous f.continuous
+
+protected theorem map_iff [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] (f : α ≃ₜ β) :
+    InnerRegular (Measure.map f μ) ↔ InnerRegular μ := by
+  refine ⟨fun h ↦ ?_, fun h ↦ h.map f⟩
+  convert h.map f.symm
+  rw [map_map f.symm.continuous.measurable f.continuous.measurable]
+  simp
 
-/-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
+end InnerRegular
+
+namespace InnerRegularCompactLTTop
+
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets,
+then any measurable set of finite measure can be approximated by a
 compact subset. See also `MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
-theorem _root_.MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A)
-    (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ K, K ⊆ A ∧ IsCompact K ∧ μ A < μ K + ε :=
-  Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
+theorem _root_.MeasurableSet.exists_isCompact_lt_add [InnerRegularCompactLTTop μ]
+    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ K, K ⊆ A ∧ IsCompact K ∧ μ A < μ K + ε :=
+  InnerRegularCompactLTTop.innerRegular.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε
+
 #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add
 
-/-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets,
+then any measurable set of finite measure can be approximated by a
 compact subset. See also `MeasurableSet.exists_isCompact_lt_add` and
 `MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
 theorem _root_.MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Space α]
-    [Regular μ] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    [InnerRegularCompactLTTop μ]  ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞)
+    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ K, K ⊆ A ∧ IsCompact K ∧ μ (A \ K) < ε := by
   rcases hA.exists_isCompact_lt_add h'A hε with ⟨K, hKA, hKc, hK⟩
   exact
@@ -513,44 +730,125 @@ theorem _root_.MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α]
         hK⟩
 #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt
 
-/-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets,
+then any measurable set of finite measure can be approximated by a
 compact subset. See also `MeasurableSet.exists_isCompact_lt_add`. -/
-theorem _root_.MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
+theorem _root_.MeasurableSet.exists_lt_isCompact_of_ne_top [InnerRegularCompactLTTop μ] ⦃A : Set α⦄
     (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) :
     ∃ K, K ⊆ A ∧ IsCompact K ∧ r < μ K :=
-  Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr
+  InnerRegularCompactLTTop.innerRegular ⟨hA, h'A⟩ _ hr
 #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top
 
-/-- Given a regular measure, any measurable set of finite mass can be approximated from
-inside by compact sets. -/
-theorem _root_.MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
-    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsCompact K), μ K :=
-  Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets,
+any measurable set of finite mass can be approximated from inside by compact sets. -/
+theorem _root_.MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [InnerRegularCompactLTTop μ]
+    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) :
+    μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsCompact K), μ K :=
+  InnerRegularCompactLTTop.innerRegular.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 
-protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β]
-    [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).Regular := by
-  haveI := OuterRegular.map f μ
-  haveI := IsFiniteMeasureOnCompacts.map μ f
-  exact
-    ⟨Regular.innerRegular.map f.toEquiv f.measurable.aemeasurable
-        (fun U hU => hU.preimage f.continuous) (fun K hK => hK.image f.continuous)
-        (fun K hK => hK.measurableSet) fun U hU => hU.measurableSet⟩
-#align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map
-
-protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).Regular := by
-  haveI := OuterRegular.smul μ hx
-  haveI := IsFiniteMeasureOnCompacts.smul μ hx
-  exact ⟨Regular.innerRegular.smul x⟩
-#align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
+/-- If `μ` is inner regular for finite measure sets with respect to compact sets, then its
+restriction to any set also is. -/
+instance restrict [h : InnerRegularCompactLTTop μ] (A : Set α) :
+    InnerRegularCompactLTTop (μ.restrict A) :=
+  ⟨InnerRegularWRT.restrict h.innerRegular A⟩
+
+instance (priority := 50) [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] :
+    InnerRegular μ := by
+  constructor
+  convert h.innerRegular with s
+  simp [measure_ne_top μ s]
+
+instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
+    [InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : WeaklyRegular μ := by
+  apply InnerRegularWRT.weaklyRegular_of_finite
+  exact InnerRegular.innerRegularWRT_isClosed_isOpen
+
+instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
+    [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : Regular μ := by
+  constructor
+  apply InnerRegularWRT.trans h.innerRegular
+  exact InnerRegularWRT.of_imp (fun U hU ↦ ⟨hU.measurableSet, measure_ne_top μ U⟩)
+
+/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
+compact space, then any compact set can be approximated from outside by open sets. -/
+protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
+    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
+    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
+    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
+  apply le_antisymm
+  · simp only [le_iInf_iff]
+    rintro U KU -
+    exact measure_mono KU
+  apply le_of_forall_lt' (fun r hr ↦ ?_)
+  simp only [iInf_lt_iff, exists_prop, exists_and_left]
+  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
+    exists_compact_between hK isOpen_univ (subset_univ _)
+  have : Fact (μ (interior L) < ∞) :=
+    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
+  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
+    apply exists_isOpen_lt_of_lt K r
+    exact (restrict_apply_le _ _).trans_lt hr
+  refine ⟨U ∩ interior L, subset_inter KU KL, U_open.inter isOpen_interior, ?_⟩
+  rwa [restrict_apply U_open.measurableSet] at hU
+
+protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
+    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
+    [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
+    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
+  have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
+    rwa [hK.measure_eq_infi_isOpen] at hr
+  simpa only [iInf_lt_iff, exists_prop, exists_and_left]
+
+protected theorem _root_.IsCompact.exists_isOpen_lt_add [InnerRegularCompactLTTop μ]
+    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
+    [BorelSpace α] {K : Set α} (hK : IsCompact K) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+     ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < μ K + ε :=
+  hK.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hK.measure_lt_top.ne hε)
+
+instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
+  by_cases hc : c = 0
+  · simp only [hc, zero_smul]
+    infer_instance
+  by_cases h'c : c = ∞
+  · constructor
+    intro s hs r hr
+    simp only [h'c, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul] at hr
+    by_cases h's : μ s = 0
+    · simp [h's] at hr
+    · simp [h'c, ENNReal.mul_eq_top, h's] at hs
+  · constructor
+    convert InnerRegularWRT.smul h.innerRegular c using 2 with s
+    have : (c • μ) s ≠ ∞ ↔ μ s ≠ ∞ := by simp [not_iff_not, ENNReal.mul_eq_top, hc, h'c]
+    simp only [this]
+
+instance smul_nnreal [InnerRegularCompactLTTop μ] (c : ℝ≥0) :
+    InnerRegularCompactLTTop (c • μ) :=
+  inferInstanceAs (InnerRegularCompactLTTop ((c : ℝ≥0∞) • μ))
+
+instance (priority := 80) [InnerRegularCompactLTTop μ] [SigmaFinite μ] : InnerRegular μ :=
+  ⟨InnerRegularCompactLTTop.innerRegular.trans InnerRegularWRT.of_sigmaFinite⟩
+
+protected theorem map_of_continuous [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] [h : InnerRegularCompactLTTop μ] {f : α → β} (hf : Continuous f) :
+    InnerRegularCompactLTTop (Measure.map f μ) := by
+  constructor
+  refine InnerRegularWRT.map h.innerRegular hf.aemeasurable ?_ (fun K hK ↦ hK.image hf) ?_
+  · rintro s ⟨hs, h's⟩
+    exact ⟨hf.measurable hs, by rwa [map_apply hf.measurable hs] at h's⟩
+  · rintro s ⟨hs, -⟩
+    exact hs
+
+end InnerRegularCompactLTTop
 
 -- Generalized and moved to another file
 #align measure_theory.measure.regular.sigma_finite MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts
 
-end Regular
-
 namespace WeaklyRegular
 
+instance zero : WeaklyRegular (0 : Measure α) :=
+  ⟨fun _ _ _r hr => ⟨∅, empty_subset _, isClosed_empty, hr⟩⟩
+
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem _root_.IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
     (hr : r < μ U) : ∃ F, F ⊆ U ∧ IsClosed F ∧ r < μ F :=
@@ -564,9 +862,8 @@ theorem _root_.IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U)
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
 
 theorem innerRegular_measurable [WeaklyRegular μ] :
-    InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
-  WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ =>
-    h₁.inter h₂.isClosed_compl
+    InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
+  WeaklyRegular.innerRegular.measurableSet_of_open (fun _ _ h₁ h₂ ↦ h₁.inter h₂.isClosed_compl)
 #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
 
 /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive
@@ -604,53 +901,143 @@ theorem _root_.MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular 
 
 /-- The restriction of a weakly regular measure to a measurable set of finite measure is
 weakly regular. -/
-theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α)
-    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) := by
+theorem restrict_of_measure_ne_top [BorelSpace α] [WeaklyRegular μ] {A : Set α}
+    (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) := by
   haveI : Fact (μ A < ∞) := ⟨h'A.lt_top⟩
-  refine' InnerRegular.weaklyRegular_of_finite (μ.restrict A) fun V V_open => _
-  simp only [restrict_apply' hA]
-  intro r hr
-  have : μ (V ∩ A) ≠ ∞ := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _)
-  rcases (V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with
-    ⟨F, hFVA, hFc, hF⟩
-  refine' ⟨F, hFVA.trans (inter_subset_left _ _), hFc, _⟩
-  rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]
-#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet
+  refine InnerRegularWRT.weaklyRegular_of_finite (μ.restrict A) (fun V V_open r hr ↦ ?_)
+  have : InnerRegularWRT (μ.restrict A) IsClosed (fun s ↦ MeasurableSet s) :=
+    InnerRegularWRT.restrict_of_measure_ne_top innerRegular_measurable h'A
+  exact this V_open.measurableSet r hr
+#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top
 
 -- see Note [lower instance priority]
-/-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type*}
-    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
+/-- Any finite measure on a metrizable space (or even a pseudo metrizable space)
+is weakly regular. -/
+instance (priority := 100) of_pseudoMetrizableSpace_of_isFiniteMeasure {X : Type*}
+    [TopologicalSpace X] [PseudoMetrizableSpace X] [MeasurableSpace X] [BorelSpace X]
+    (μ : Measure X) [IsFiniteMeasure μ] :
     WeaklyRegular μ :=
-  (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
+  (InnerRegularWRT.of_pseudoMetrizableSpace μ).weaklyRegular_of_finite μ
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_of_isFiniteMeasure
 
 -- see Note [lower instance priority]
-/-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
-is weakly regular. -/
-instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type*}
-    [PseudoEMetricSpace X] [SecondCountableTopology X] [MeasurableSpace X]
+/-- Any locally finite measure on a second countable metrizable space
+(or even a pseudo metrizable space) is weakly regular. -/
+instance (priority := 100) of_pseudoMetrizableSpace_secondCountable_of_locallyFinite {X : Type*}
+    [TopologicalSpace X] [PseudoMetrizableSpace X] [SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
-  haveI : OuterRegular μ := by
+  have : OuterRegular μ := by
     refine' (μ.finiteSpanningSetsInOpen'.mono' fun U hU => _).outerRegular
     have : Fact (μ U < ∞) := ⟨hU.2⟩
     exact ⟨hU.1, inferInstance⟩
-  ⟨InnerRegular.of_pseudoEMetricSpace μ⟩
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite
+  ⟨InnerRegularWRT.of_pseudoMetrizableSpace μ⟩
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoMetrizableSpace_secondCountable_of_locallyFinite
+
+protected theorem smul [WeaklyRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).WeaklyRegular := by
+  haveI := OuterRegular.smul μ hx
+  exact ⟨WeaklyRegular.innerRegular.smul x⟩
+
+instance smul_nnreal [WeaklyRegular μ] (c : ℝ≥0) : WeaklyRegular (c • μ) :=
+  WeaklyRegular.smul coe_ne_top
 
 end WeaklyRegular
 
-attribute [local instance] EMetric.secondCountable_of_sigmaCompact
+namespace Regular
+
+instance zero : Regular (0 : Measure α) :=
+  ⟨fun _ _ _r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩
+#align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero
+
+/-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/
+theorem _root_.IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
+    (hr : r < μ U) : ∃ K, K ⊆ U ∧ IsCompact K ∧ r < μ K :=
+  Regular.innerRegular hU r hr
+#align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact
+
+/-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
+theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
+    [Regular μ] : μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K :=
+  Regular.innerRegular.measure_eq_iSup hU
+#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
+
+theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
+  simp_rw [Ne.def, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
+    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+#align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null
+
+/-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
+compact subset. See also `MeasurableSet.exists_isCompact_lt_add` and
+`MeasurableSet.exists_lt_isCompact_of_ne_top`. -/
+instance (priority := 100) [Regular μ] : InnerRegularCompactLTTop μ :=
+  ⟨Regular.innerRegular.measurableSet_of_open (fun _ _ hs hU ↦ hs.diff hU)⟩
+#noalign measure_theory.measure.regular.inner_regular_measurable
+
+protected theorem map [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).Regular := by
+  haveI := OuterRegular.map f μ
+  haveI := IsFiniteMeasureOnCompacts.map μ f
+  exact
+    ⟨Regular.innerRegular.map' f.toMeasurableEquiv
+        (fun U hU => hU.preimage f.continuous)
+        (fun K hK => hK.image f.continuous)⟩
+#align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map
+
+protected theorem map_iff [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
+    [BorelSpace β] (f : α ≃ₜ β) :
+    Regular (Measure.map f μ) ↔ Regular μ := by
+  refine ⟨fun h ↦ ?_, fun h ↦ h.map f⟩
+  convert h.map f.symm
+  rw [map_map f.symm.continuous.measurable f.continuous.measurable]
+  simp
+
+protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).Regular := by
+  haveI := OuterRegular.smul μ hx
+  haveI := IsFiniteMeasureOnCompacts.smul μ hx
+  exact ⟨Regular.innerRegular.smul x⟩
+#align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
+
+instance smul_nnreal [Regular μ] (c : ℝ≥0) : Regular (c • μ) := Regular.smul coe_ne_top
+
+/-- The restriction of a regular measure to a set of finite measure is regular. -/
+theorem restrict_of_measure_ne_top [ClosableCompactSubsetOpenSpace α] [BorelSpace α] [Regular μ]
+    {A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A) := by
+  have : WeaklyRegular (μ.restrict A) := WeaklyRegular.restrict_of_measure_ne_top h'A
+  constructor
+  intro V hV r hr
+  have R : restrict μ A V ≠ ∞ := by
+    rw [restrict_apply hV.measurableSet]
+    exact ((measure_mono (inter_subset_right _ _)).trans_lt h'A.lt_top).ne
+  exact MeasurableSet.exists_lt_isCompact_of_ne_top hV.measurableSet R hr
+
+end Regular
 
 -- see Note [lower instance priority]
-/-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
+/-- Any locally finite measure on a `σ`-compact pseudometrizable space is regular. -/
 instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type*}
-    [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
-    [IsLocallyFiniteMeasure μ] : Regular μ where
-  lt_top_of_isCompact _K hK := hK.measure_lt_top
-  innerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
+    [TopologicalSpace X] [PseudoMetrizableSpace X] [SigmaCompactSpace X] [MeasurableSpace X]
+    [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : Regular μ := by
+  let A : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
+  exact ⟨(InnerRegularWRT.isCompact_isClosed μ).trans (InnerRegularWRT.of_pseudoMetrizableSpace μ)⟩
 #align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
 
+/-- Any sigma finite measure on a `σ`-compact pseudometrizable space is inner regular. -/
+instance (priority := 100) {X : Type*}
+    [TopologicalSpace X] [PseudoMetrizableSpace X] [SigmaCompactSpace X] [MeasurableSpace X]
+    [BorelSpace X] (μ : Measure X) [SigmaFinite μ] : InnerRegular μ := by
+  refine ⟨(InnerRegularWRT.isCompact_isClosed μ).trans ?_⟩
+  refine InnerRegularWRT.of_restrict (fun n ↦ ?_)
+    (univ_subset_iff.2 (iUnion_spanningSets μ)) (monotone_spanningSets μ)
+  have : Fact (μ (spanningSets μ n) < ∞) := ⟨measure_spanningSets_lt_top μ n⟩
+  exact WeaklyRegular.innerRegular_measurable.trans InnerRegularWRT.of_sigmaFinite
+
+/- Check that typeclass inference works to guarantee regularity and inner regularity in
+interesting situations. -/
+example [LocallyCompactSpace α] [RegularSpace α] [BorelSpace α] [SecondCountableTopology α]
+    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] : Regular μ := by infer_instance
+
+example [LocallyCompactSpace α] [RegularSpace α] [BorelSpace α] [SecondCountableTopology α]
+    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] : InnerRegular μ := by infer_instance
+
 end Measure
 
 end MeasureTheory

All the other properties of topological spaces like T0Space or RegularSpace are in the root namespace. Many files were opening TopologicalSpace just for the sake of shortening TopologicalSpace.SecondCountableTopology...

@@ -629,7 +629,7 @@ instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type*}
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
 instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type*}
-    [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
+    [PseudoEMetricSpace X] [SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : OuterRegular μ := by
     refine' (μ.finiteSpanningSetsInOpen'.mono' fun U hU => _).outerRegular

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -326,7 +326,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
       rw [H₁]
       exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne
-    rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
+    rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
     rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
     exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
   choose U hAU hUo hU using this
@@ -411,7 +411,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     have : Tendsto (fun t => (∑ k in t, μ (s k)) + ε / 2) atTop (𝓝 <| μ (⋃ n, s n) + ε / 2) := by
       rw [measure_iUnion hsd hsm]
       exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds
-    rcases(this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
+    rcases (this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
     -- the approximating open set is constructed by taking for each `s n` an approximating open set
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
     refine'
@@ -611,7 +611,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
   simp only [restrict_apply' hA]
   intro r hr
   have : μ (V ∩ A) ≠ ∞ := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _)
-  rcases(V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with
+  rcases (V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with
     ⟨F, hFVA, hFc, hF⟩
   refine' ⟨F, hFVA.trans (inter_subset_left _ _), hFc, _⟩
   rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]

Generalize MeasureTheory.Measure.Regular.sigmaFinite from a regular measure to a measure finite on compacts, rename it to MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts.

@@ -544,14 +544,8 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • 
   exact ⟨Regular.innerRegular.smul x⟩
 #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
 
--- see Note [lower instance priority]
-/-- A regular measure in a σ-compact space is σ-finite. -/
-instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : SigmaFinite μ :=
-  ⟨⟨{   set := compactCovering α
-        set_mem := fun _ => trivial
-        finite := fun n => (isCompact_compactCovering α n).measure_lt_top
-        spanning := iUnion_compactCovering α }⟩⟩
-#align measure_theory.measure.regular.sigma_finite MeasureTheory.Measure.Regular.sigmaFinite
+-- Generalized and moved to another file
+#align measure_theory.measure.regular.sigma_finite MeasureTheory.SigmaFinite.of_isFiniteMeasureOnCompacts
 
 end Regular
 

We define Alexandrov-discrete spaces as topological spaces where the intersection of a family of open sets is open.

This PR only gives a minimal API because the goal is to ensure that lemma names like isOpen_sInter are free to use for AlexandrovDiscrete. The existing lemmas are getting prefixed by Set.Finite or suffixed by _of_finite.

@@ -416,7 +416,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
     refine'
       ⟨⋃ k ∈ t, F k, iUnion_mono fun k => iUnion_subset fun _ => hFs _, ⋃ n, U n, iUnion_mono hsU,
-        isClosed_biUnion t.finite_toSet fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
+        isClosed_biUnion_finset fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
     · calc
         (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 := by
           rw [← sum_add_distrib]

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -147,7 +147,7 @@ def InnerRegular {α} {_ : MeasurableSpace α} (μ : Measure α) (p q : Set α 
 
 namespace InnerRegular
 
-variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
+variable {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α}
   {ε : ℝ≥0∞}
 
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
@@ -193,7 +193,7 @@ theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegula
 
 end InnerRegular
 
-variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
+variable {α β : Type*} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α}
 
 /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`.
 
@@ -436,7 +436,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
 
 /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside
 by closed sets. -/
-theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X]
+theorem of_pseudoEMetricSpace {X : Type*} [PseudoEMetricSpace X] [MeasurableSpace X]
     (μ : Measure X) : InnerRegular μ IsClosed IsOpen := by
   intro U hU r hr
   rcases hU.exists_iUnion_isClosed with ⟨F, F_closed, -, rfl, F_mono⟩
@@ -446,7 +446,7 @@ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpa
 #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace
 
 /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/
-theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X]
+theorem isCompact_isClosed {X : Type*} [TopologicalSpace X] [SigmaCompactSpace X]
     [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed := by
   intro F hF r hr
   set B : ℕ → Set X := compactCovering X
@@ -625,7 +625,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
 
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
+instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type*}
     [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
     WeaklyRegular μ :=
   (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
@@ -634,7 +634,7 @@ instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
-instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _}
+instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type*}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : OuterRegular μ := by
@@ -650,7 +650,7 @@ attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
-instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type _}
+instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type*}
     [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
     [IsLocallyFiniteMeasure μ] : Regular μ where
   lt_top_of_isCompact _K hK := hK.measure_lt_top

@@ -344,7 +344,7 @@ namespace InnerRegular
 variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞}
 
 /-- If a measure is inner regular (using closed or compact sets), then every measurable set of
-finite measure can by approximated by a (closed or compact) subset. -/
+finite measure can be approximated by a (closed or compact) subset. -/
 theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (h0 : p ∅)
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
     InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ := by

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris Van Doorn, Yury Kudryashov
-
-! This file was ported from Lean 3 source module measure_theory.measure.regular
-! leanprover-community/mathlib commit bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
 
+#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
+
 /-!
 # Regular measures
 

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -336,8 +336,8 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
   refine' ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _⟩
   calc
     μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_iUnion_le _
-    _ ≤ ∑' n, μ (A n) + δ n := (ENNReal.tsum_le_tsum fun n => (hU n).le)
-    _ = (∑' n, μ (A n)) + ∑' n, δ n := ENNReal.tsum_add
+    _ ≤ ∑' n, (μ (A n) + δ n) := (ENNReal.tsum_le_tsum fun n => (hU n).le)
+    _ = ∑' n, μ (A n) + ∑' n, δ n := ENNReal.tsum_add
     _ = μ (⋃ n, A n) + ∑' n, δ n := (congr_arg₂ (· + ·) (measure_iUnion hAd hAm).symm rfl)
     _ < r := hδε
 #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
@@ -432,7 +432,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
             fun k _ => (hFc k).measurableSet]
     · calc
         μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_iUnion_le _
-        _ ≤ ∑' n, μ (s n) + δ n := (ENNReal.tsum_le_tsum hU)
+        _ ≤ ∑' n, (μ (s n) + δ n) := (ENNReal.tsum_le_tsum hU)
         _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]
         _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _
 #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite

@@ -454,7 +454,7 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace
   intro F hF r hr
   set B : ℕ → Set X := compactCovering X
   have hBc : ∀ n, IsCompact (F ∩ B n) := fun n => (isCompact_compactCovering X n).inter_left hF
-  have hBU : (⋃ n, F ∩ B n) = F := by rw [← inter_iUnion, iUnion_compactCovering, Set.inter_univ]
+  have hBU : ⋃ n, F ∩ B n = F := by rw [← inter_iUnion, iUnion_compactCovering, Set.inter_univ]
   have : μ F = ⨆ n, μ (F ∩ B n) := by
     rw [← measure_iUnion_eq_iSup, hBU]
     exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)

@@ -220,7 +220,7 @@ class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegul
 /-- A measure `μ` is weakly regular if
   - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable;
   - it is inner regular for open sets, using closed sets:
-    `μ(U) = sup {μ(F) | F ⊆ U compact}` for `U` open. -/
+    `μ(U) = sup {μ(F) | F ⊆ U closed}` for `U` open. -/
 class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
   protected innerRegular : InnerRegular μ IsClosed IsOpen
 #align measure_theory.measure.weakly_regular MeasureTheory.Measure.WeaklyRegular

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -323,7 +323,7 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {
     rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]
   rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
   rw [lt_tsub_iff_right, add_comm] at hδε
-  have : ∀ n, ∃ (U : _)(_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
+  have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
     intro n
     have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
     have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
@@ -352,7 +352,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
     InnerRegular μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ := by
   rintro s ⟨hs, hμs⟩ r hr
-  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
+  obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _) (_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by
     use (μ s - r) / 2
     simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩

@@ -154,7 +154,7 @@ variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α
   {ε : ℝ≥0∞}
 
 theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) :
-    μ U = ⨆ (K) (_h : K ⊆ U) (_hK : p K), μ K := by
+    μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K := by
   refine'
     le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
   simpa only [lt_iSup_iff, exists_prop] using H hU r hr
@@ -253,7 +253,7 @@ theorem _root_.Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : 
 /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets
 containing it. -/
 theorem _root_.Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
-    μ A = ⨅ (U : Set α) (_h : A ⊆ U) (_h2 : IsOpen U), μ U := by
+    μ A = ⨅ (U : Set α) (_ : A ⊆ U) (_ : IsOpen U), μ U := by
   refine' le_antisymm (le_iInf₂ fun s hs => le_iInf fun _ => μ.mono hs) _
   refine' le_of_forall_lt' fun r hr => _
   simpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr
@@ -479,7 +479,7 @@ theorem _root_.IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : Is
 
 /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
 theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
-    [Regular μ] : μ U = ⨆ (K : Set α) (_h : K ⊆ U) (_h2 : IsCompact K), μ K :=
+    [Regular μ] : μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K :=
   Regular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact
 
@@ -527,7 +527,7 @@ theorem _root_.MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : S
 /-- Given a regular measure, any measurable set of finite mass can be approximated from
 inside by compact sets. -/
 theorem _root_.MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Set α⦄
-    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_h : K ⊆ A) (_h : IsCompact K), μ K :=
+    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsCompact K), μ K :=
   Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
 
@@ -568,7 +568,7 @@ theorem _root_.IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU
 
 /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/
 theorem _root_.IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
-    [WeaklyRegular μ] : μ U = ⨆ (F) (_h : F ⊆ U) (_h : IsClosed F), μ F :=
+    [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (_ : IsClosed F), μ F :=
   WeaklyRegular.innerRegular.measure_eq_iSup hU
 #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed
 
@@ -607,7 +607,7 @@ theorem _root_.MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
 inside by closed sets. -/
 theorem _root_.MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
-    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_h : K ⊆ A) (_h : IsClosed K), μ K :=
+    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsClosed K), μ K :=
   innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩
 #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
 

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.

@@ -86,7 +86,7 @@ is automatically satisfied by any finite measure on a metric space.
 *  `MeasurableSet.exists_lt_isClosed_of_ne_top` and `MeasurableSet.exists_isClosed_lt_add`:
   a measurable set of finite measure can be approximated by a closed subset (stated as
   `r < μ F` and `μ s < μ F + ε`, respectively).
-* `MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure` is an
+* `MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure` is an
   instance registering that a finite measure on a metric space is weakly regular (in fact, a pseudo
   emetric space is enough);
 * `MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite`
@@ -104,7 +104,7 @@ is automatically satisfied by any finite measure on a metric space.
 *  `MeasurableSet.exists_lt_isCompact_of_ne_top` and `MeasurableSet.exists_isCompact_lt_add`:
   a measurable set of finite measure can be approximated by a compact subset (stated as
   `r < μ K` and `μ s < μ K + ε`, respectively).
-* `MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure` is an
+* `MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure` is an
   instance registering that a locally finite measure on a `σ`-compact metric space is regular (in
   fact, an emetric space is enough).
 
@@ -213,7 +213,7 @@ class OuterRegular (μ : Measure α) : Prop where
   - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable;
   - it is inner regular for open sets, using compact sets:
     `μ(U) = sup {μ(K) | K ⊆ U compact}` for `U` open. -/
-class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
+class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
   innerRegular : InnerRegular μ IsCompact IsOpen
 #align measure_theory.measure.regular MeasureTheory.Measure.Regular
 
@@ -374,7 +374,7 @@ open Finset
 
 /-- In a finite measure space, assume that any open set can be approximated from inside by closed
 sets. Then the measure is weakly regular. -/
-theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure μ]
+theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
     (H : InnerRegular μ IsClosed IsOpen) : WeaklyRegular μ := by
   have hfin : ∀ {s}, μ s ≠ ⊤ := @(measure_ne_top μ)
   suffices ∀ s, MeasurableSet s → ∀ ε, ε ≠ 0 → ∃ F, F ⊆ s ∧ ∃ U, U ⊇ s ∧
@@ -543,7 +543,7 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica
 
 protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).Regular := by
   haveI := OuterRegular.smul μ hx
-  haveI := FiniteMeasureOnCompacts.smul μ hx
+  haveI := IsFiniteMeasureOnCompacts.smul μ hx
   exact ⟨Regular.innerRegular.smul x⟩
 #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul
 
@@ -628,18 +628,18 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α
 
 -- see Note [lower instance priority]
 /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/
-instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _}
-    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [FiniteMeasure μ] :
+instance (priority := 100) of_pseudoEMetricSpace_of_isFiniteMeasure {X : Type _}
+    [PseudoEMetricSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] :
     WeaklyRegular μ :=
   (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ
-#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure
+#align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_isFiniteMeasure
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space)
 is weakly regular. -/
 instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _}
     [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X]
-    [BorelSpace X] (μ : Measure X) [LocallyFiniteMeasure μ] : WeaklyRegular μ :=
+    [BorelSpace X] (μ : Measure X) [IsLocallyFiniteMeasure μ] : WeaklyRegular μ :=
   haveI : OuterRegular μ := by
     refine' (μ.finiteSpanningSetsInOpen'.mono' fun U hU => _).outerRegular
     have : Fact (μ U < ∞) := ⟨hU.2⟩
@@ -653,12 +653,12 @@ attribute [local instance] EMetric.secondCountable_of_sigmaCompact
 
 -- see Note [lower instance priority]
 /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/
-instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure {X : Type _}
+instance (priority := 100) Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure {X : Type _}
     [EMetricSpace X] [SigmaCompactSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X)
-    [LocallyFiniteMeasure μ] : Regular μ where
+    [IsLocallyFiniteMeasure μ] : Regular μ where
   lt_top_of_isCompact _K hK := hK.measure_lt_top
   innerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ)
-#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure
+#align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure
 
 end Measure
 

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -132,9 +132,7 @@ proofs or statements do not apply directly.
 -/
 
 
-open Set Filter
-
-open ENNReal Topology NNReal BigOperators
+open Set Filter ENNReal Topology NNReal BigOperators
 
 namespace MeasureTheory
 
@@ -358,7 +356,7 @@ theorem measurableSet_of_open [OuterRegular μ] (H : InnerRegular μ p IsOpen) (
     use (μ s - r) / 2
     simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]
   rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩
-  rcases(U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
+  rcases (U \ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩
   replace hsU' := diff_subset_comm.1 hsU'
   rcases H.exists_subset_lt_add h0 hUo hUt.ne hε with ⟨K, hKU, hKc, hKr⟩
   refine' ⟨K \ U', fun x hx => hsU' ⟨hKU hx.1, hx.2⟩, hd hKc hU'o, ENNReal.sub_lt_of_lt_add hεs _⟩

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -140,7 +140,7 @@ namespace MeasureTheory
 
 namespace Measure
 
-/-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`,
+/-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : Set α → Prop`,
 if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K`
 of measure greater than `r`.
 

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com>