measure_theory.function.continuous_map_dense ⟷ Mathlib.MeasureTheory.Function.ContinuousMapDense

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

@@ -146,7 +146,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
 #align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le /-
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
@@ -255,7 +255,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le /-
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/

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

@@ -112,7 +112,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     by
     intro x
     by_cases hv : x ∈ v
-    · rw [← Set.diff_union_of_subset hsv] at hv 
+    · rw [← Set.diff_union_of_subset hsv] at hv
       cases' hv with hsv hs
       ·
         simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
@@ -221,11 +221,11 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
   have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_has_compact_support_snorm_sub_le ENNReal.coe_ne_top A with
     ⟨g, g_support, hg, g_cont, g_mem⟩
-  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg 
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
   refine' ⟨g, g_support, _, g_cont, g_mem⟩
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
-    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
   exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 -/
@@ -325,11 +325,11 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
     simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
   have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_bounded_continuous_snorm_sub_le ENNReal.coe_ne_top A with ⟨g, hg, g_mem⟩
-  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg 
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
   refine' ⟨g, _, g_mem⟩
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
-    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
   exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
 -/

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

@@ -226,7 +226,7 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
-  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 -/
 
@@ -330,7 +330,7 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
-  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
 -/
 

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

@@ -98,7 +98,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
   have hsv : s ⊆ v := subset_inter hsu sV
   have hμv : μ v < ∞ := (measure_mono (inter_subset_right _ _)).trans_lt h'V
   obtain ⟨g, hgv, hgs, hg_range⟩ :=
-    exists_continuous_zero_one_of_closed (u_open.inter V_open).isClosed_compl s_closed
+    exists_continuous_zero_one_of_isClosed (u_open.inter V_open).isClosed_compl s_closed
       (disjoint_compl_left_iff.2 hsv)
   -- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
   -- that this is pointwise bounded by the indicator of the set `v \ s`, which has small measure.

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
-import Mathbin.MeasureTheory.Measure.Regular
-import Mathbin.MeasureTheory.Function.SimpleFuncDenseLp
-import Mathbin.Topology.UrysohnsLemma
-import Mathbin.MeasureTheory.Integral.Bochner
+import MeasureTheory.Measure.Regular
+import MeasureTheory.Function.SimpleFuncDenseLp
+import Topology.UrysohnsLemma
+import MeasureTheory.Integral.Bochner
 
 #align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"
 
@@ -146,7 +146,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
 #align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le /-
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
@@ -255,7 +255,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le /-
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/

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

@@ -264,9 +264,9 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
     ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ε ∧ Memℒp g p μ :=
   by
   suffices H :
-    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ mem_ℒp g p μ ∧ Metric.Bounded (range g)
+    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ mem_ℒp g p μ ∧ Bornology.IsBounded (range g)
   · rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
-    exact ⟨⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩, hg, g_mem⟩
+    exact ⟨⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩, hg, g_mem⟩
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.
   -- First check the latter easy facts.
@@ -275,9 +275,9 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩
     refine' ⟨f_cont.add g_cont, f_mem.add g_mem, _⟩
-    let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.bounded_range_iff.1 f_bd⟩
-    let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩
-    exact (f' + g').bounded_range
+    let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.isBounded_range_iff.1 f_bd⟩
+    let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩
+    exact (f' + g').isBounded_range
   -- ae strong measurability
   · exact fun f ⟨_, h, _⟩ => h.AEStronglyMeasurable
   -- We are left with approximating characteristic functions.
@@ -309,7 +309,7 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
           I2 I1).le
     simp only [sub_add_sub_cancel]
   refine' ⟨f, I3, f_cont, f_mem, _⟩
-  exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
+  exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).isBounded_range
 #align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le
 -/
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Measure.Regular
 import Mathbin.MeasureTheory.Function.SimpleFuncDenseLp
 import Mathbin.Topology.UrysohnsLemma
 import Mathbin.MeasureTheory.Integral.Bochner
 
+#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"
+
 /-!
 # Approximation in Lᵖ by continuous functions
 
@@ -149,7 +146,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
 #align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le /-
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
@@ -258,7 +255,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 #print MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le /-
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/

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

@@ -77,6 +77,7 @@ namespace MeasureTheory
 
 variable [NormedSpace ℝ E]
 
+#print MeasureTheory.exists_continuous_snorm_sub_le_of_closed /-
 /-- A variant of Urysohn's lemma, `ℒ^p` version, for an outer regular measure `μ`:
 consider two sets `s ⊆ u` which are respectively closed and open with `μ s < ∞`, and a vector `c`.
 Then one may find a continuous function `f` equal to `c` on `s` and to `0` outside of `u`,
@@ -146,8 +147,10 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
       gc_support.trans (inter_subset_left _ _), gc_mem⟩
   exact hη _ ((measure_mono (diff_subset_diff (inter_subset_right _ _) subset.rfl)).trans hV.le)
 #align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+#print MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le /-
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
 theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [μ.regular] (hp : p ≠ ∞)
@@ -204,7 +207,9 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   contrapose! hx
   exact interior_subset (f_support hx)
 #align measure_theory.mem_ℒp.exists_has_compact_support_snorm_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le
+-/
 
+#print MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le /-
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `0 < p < ∞`, version in terms of `∫`. -/
 theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpace α] [μ.regular]
@@ -226,7 +231,9 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
   exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
+-/
 
+#print MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le /-
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
 continuous functions, version in terms of `∫⁻`. -/
 theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpace α] [μ.regular]
@@ -236,7 +243,9 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpac
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_has_compact_support_lintegral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
+-/
 
+#print MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le /-
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
 continuous functions, version in terms of `∫`. -/
 theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace α] [μ.regular]
@@ -247,8 +256,10 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
     hf ⊢
   simpa using hf.exists_has_compact_support_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+#print MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le /-
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/
 theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
@@ -303,7 +314,9 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
   refine' ⟨f, I3, f_cont, f_mem, _⟩
   exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
 #align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le
+-/
 
+#print MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le /-
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `0 < p < ∞`,
 version in terms of `∫`. -/
 theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
@@ -322,7 +335,9 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
   exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
+-/
 
+#print MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le /-
 /-- Any integrable function can be approximated by bounded continuous functions,
 version in terms of `∫⁻`. -/
 theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
@@ -332,7 +347,9 @@ theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular]
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le
+-/
 
+#print MeasureTheory.Integrable.exists_boundedContinuous_integral_sub_le /-
 /-- Any integrable function can be approximated by bounded continuous functions,
 version in terms of `∫`. -/
 theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
@@ -343,11 +360,13 @@ theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {
     hf ⊢
   simpa using hf.exists_bounded_continuous_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_boundedContinuous_integral_sub_le
+-/
 
 namespace Lp
 
 variable (E)
 
+#print MeasureTheory.Lp.boundedContinuousFunction_dense /-
 /-- A function in `Lp` can be approximated in `Lp` by continuous functions. -/
 theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)]
     (hp : p ≠ ∞) [μ.WeaklyRegular] : (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=
@@ -368,6 +387,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   filter_upwards [coe_fn_sub f (g_mem.to_Lp g), g_mem.coe_fn_to_Lp] with x hx h'x
   simp only [hx, Pi.sub_apply, sub_right_inj, h'x]
 #align measure_theory.Lp.bounded_continuous_function_dense MeasureTheory.Lp.boundedContinuousFunction_dense
+-/
 
 end Lp
 
@@ -377,12 +397,11 @@ variable [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠
 
 variable (𝕜 : Type _) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E]
 
-include _i hp
-
 variable (E) (μ)
 
 namespace BoundedContinuousFunction
 
+#print BoundedContinuousFunction.toLp_denseRange /-
 theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ) :=
   by
@@ -392,11 +411,13 @@ theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     by exact congr_arg coe this
   simp [range_to_Lp p μ, MeasureTheory.Lp.boundedContinuousFunction_dense E hp]
 #align bounded_continuous_function.to_Lp_dense_range BoundedContinuousFunction.toLp_denseRange
+-/
 
 end BoundedContinuousFunction
 
 namespace ContinuousMap
 
+#print ContinuousMap.toLp_denseRange /-
 theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) :=
   by
@@ -406,6 +427,7 @@ theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ
     by exact congr_arg coe this
   simp [range_to_Lp p μ, MeasureTheory.Lp.boundedContinuousFunction_dense E hp]
 #align continuous_map.to_Lp_dense_range ContinuousMap.toLp_denseRange
+-/
 
 end ContinuousMap
 

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

@@ -211,7 +211,7 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
     {p : ℝ} (hp : 0 < p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α → E,
       HasCompactSupport g ∧
-        (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ :=
+        ∫ x, ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ :=
   by
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
@@ -231,8 +231,7 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
 continuous functions, version in terms of `∫⁻`. -/
 theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpace α] [μ.regular]
     {f : α → E} (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ g : α → E,
-      HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
+    ∃ g : α → E, HasCompactSupport g ∧ ∫⁻ x, ‖f x - g x‖₊ ∂μ ≤ ε ∧ Continuous g ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top hε
@@ -242,7 +241,7 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpac
 continuous functions, version in terms of `∫`. -/
 theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace α] [μ.regular]
     {f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
-    ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
+    ∃ g : α → E, HasCompactSupport g ∧ ∫ x, ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
     hf ⊢
@@ -309,7 +308,7 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
 version in terms of `∫`. -/
 theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
     {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
-    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ :=
+    ∃ g : α →ᵇ E, ∫ x, ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ :=
   by
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
@@ -328,7 +327,7 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
 version in terms of `∫⁻`. -/
 theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ :=
+    ∃ g : α →ᵇ E, ∫⁻ x, ‖f x - g x‖₊ ∂μ ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top hε
@@ -338,7 +337,7 @@ theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular]
 version in terms of `∫`. -/
 theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
-    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ :=
+    ∃ g : α →ᵇ E, ∫ x, ‖f x - g x‖ ∂μ ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
     hf ⊢

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit e0736bb5b48bdadbca19dbd857e12bee38ccfbb8
+! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.MeasureTheory.Integral.Bochner
 /-!
 # Approximation in Lᵖ by continuous functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves that bounded continuous functions are dense in `Lp E p μ`, for `p < ∞`, if the
 domain `α` of the functions is a normal topological space and the measure `μ` is weakly regular.
 It also proves the same results for approximation by continuous functions with compact support

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

@@ -144,7 +144,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
   exact hη _ ((measure_mono (diff_subset_diff (inter_subset_right _ _) subset.rfl)).trans hV.le)
 #align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
 theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [μ.regular] (hp : p ≠ ∞)
@@ -246,7 +246,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
   simpa using hf.exists_has_compact_support_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/
 theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

@@ -249,7 +249,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/
-theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
+theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
     (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ε ∧ Memℒp g p μ :=
   by
@@ -300,11 +300,11 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
     simp only [sub_add_sub_cancel]
   refine' ⟨f, I3, f_cont, f_mem, _⟩
   exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
-#align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_snorm_sub_le
+#align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le
 
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `0 < p < ∞`,
 version in terms of `∫`. -/
-theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
+theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
     {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ :=
   by
@@ -319,28 +319,28 @@ theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
   exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
-#align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_integral_rpow_sub_le
+#align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,
 version in terms of `∫⁻`. -/
-theorem Integrable.exists_bounded_continuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
+theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top hε
-#align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_lintegral_sub_le
+#align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,
 version in terms of `∫`. -/
-theorem Integrable.exists_bounded_continuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
+theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
     hf ⊢
   simpa using hf.exists_bounded_continuous_integral_rpow_sub_le zero_lt_one hε
-#align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_integral_sub_le
+#align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_boundedContinuous_integral_sub_le
 
 namespace Lp
 
@@ -357,7 +357,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   intro ε hε
   have A : ENNReal.ofReal ε ≠ 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, hε]
   obtain ⟨g, hg, g_mem⟩ : ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ENNReal.ofReal ε ∧ mem_ℒp g p μ
-  exact (Lp.mem_ℒp f).exists_bounded_continuous_snorm_sub_le hp A
+  exact (Lp.mem_ℒp f).exists_boundedContinuous_snorm_sub_le hp A
   refine' ⟨g_mem.to_Lp _, _, ⟨g, rfl⟩⟩
   simp only [dist_eq_norm, Metric.mem_closedBall']
   rw [Lp.norm_def]

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

@@ -188,7 +188,8 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
     ⟨f, f_cont, I2, f_bound, f_support, f_mem⟩
   have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
     by
-    convert(hδ _ _
+    convert
+      (hδ _ _
           (f_mem.ae_strongly_measurable.sub
             (ae_strongly_measurable_const.indicator s_compact.measurable_set))
           ((ae_strongly_measurable_const.indicator s_compact.measurable_set).sub
@@ -289,7 +290,8 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
     ⟨f, f_cont, I2, f_bound, -, f_mem⟩
   have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
     by
-    convert(hδ _ _
+    convert
+      (hδ _ _
           (f_mem.ae_strongly_measurable.sub
             (ae_strongly_measurable_const.indicator s_closed.measurable_set))
           ((ae_strongly_measurable_const.indicator s_closed.measurable_set).sub
@@ -361,7 +363,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   rw [Lp.norm_def]
   convert ENNReal.toReal_le_of_le_ofReal hε.le hg using 2
   apply snorm_congr_ae
-  filter_upwards [coe_fn_sub f (g_mem.to_Lp g), g_mem.coe_fn_to_Lp]with x hx h'x
+  filter_upwards [coe_fn_sub f (g_mem.to_Lp g), g_mem.coe_fn_to_Lp] with x hx h'x
   simp only [hx, Pi.sub_apply, sub_right_inj, h'x]
 #align measure_theory.Lp.bounded_continuous_function_dense MeasureTheory.Lp.boundedContinuousFunction_dense
 
@@ -379,7 +381,7 @@ variable (E) (μ)
 
 namespace BoundedContinuousFunction
 
-theorem toLp_denseRange [μ.WeaklyRegular] [FiniteMeasure μ] :
+theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
@@ -393,7 +395,7 @@ end BoundedContinuousFunction
 
 namespace ContinuousMap
 
-theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [FiniteMeasure μ] :
+theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

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

@@ -91,7 +91,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε
   exact exists_snorm_indicator_le hp c hε
   have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
-  obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α)(H : V ⊇ s), IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η
+  obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α) (H : V ⊇ s), IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η
   exact s_closed.measurable_set.exists_is_open_diff_lt hs ηpos.ne'
   let v := u ∩ V
   have hsv : s ⊆ v := subset_inter hsu sV
@@ -111,7 +111,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     by
     intro x
     by_cases hv : x ∈ v
-    · rw [← Set.diff_union_of_subset hsv] at hv
+    · rw [← Set.diff_union_of_subset hsv] at hv 
       cases' hv with hsv hs
       ·
         simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
@@ -174,7 +174,7 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
     ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ δ
   exact exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
-  obtain ⟨s, st, s_compact, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsCompact s ∧ μ (t \ s) < η
+  obtain ⟨s, st, s_compact, μs⟩ : ∃ (s : _) (_ : s ⊆ t), IsCompact s ∧ μ (t \ s) < η
   exact ht.exists_is_compact_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
   have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ δ :=
@@ -215,11 +215,11 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
   have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_has_compact_support_snorm_sub_le ENNReal.coe_ne_top A with
     ⟨g, g_support, hg, g_cont, g_mem⟩
-  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg 
   refine' ⟨g, g_support, _, g_cont, g_mem⟩
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
-    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
   exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 
@@ -230,7 +230,7 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpac
     ∃ g : α → E,
       HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
   by
-  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_has_compact_support_lintegral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
 
@@ -241,7 +241,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
     ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
-    hf⊢
+    hf ⊢
   simpa using hf.exists_has_compact_support_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 
@@ -277,7 +277,7 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
     ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ δ
   exact exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
-  obtain ⟨s, st, s_closed, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsClosed s ∧ μ (t \ s) < η
+  obtain ⟨s, st, s_closed, μs⟩ : ∃ (s : _) (_ : s ⊆ t), IsClosed s ∧ μ (t \ s) < η
   exact ht.exists_is_closed_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
   have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ δ :=
@@ -311,11 +311,11 @@ theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular
     simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
   have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_bounded_continuous_snorm_sub_le ENNReal.coe_ne_top A with ⟨g, hg, g_mem⟩
-  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg 
   refine' ⟨g, _, g_mem⟩
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
-    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg 
   exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_integral_rpow_sub_le
 
@@ -325,7 +325,7 @@ theorem Integrable.exists_bounded_continuous_lintegral_sub_le [μ.WeaklyRegular]
     (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ :=
   by
-  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢
   exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_lintegral_sub_le
 
@@ -336,7 +336,7 @@ theorem Integrable.exists_bounded_continuous_integral_sub_le [μ.WeaklyRegular]
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
-    hf⊢
+    hf ⊢
   simpa using hf.exists_bounded_continuous_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_integral_sub_le
 

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

@@ -363,7 +363,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   apply snorm_congr_ae
   filter_upwards [coe_fn_sub f (g_mem.to_Lp g), g_mem.coe_fn_to_Lp]with x hx h'x
   simp only [hx, Pi.sub_apply, sub_right_inj, h'x]
-#align measure_theory.Lp.bounded_continuous_function_dense MeasureTheory.lp.boundedContinuousFunction_dense
+#align measure_theory.Lp.bounded_continuous_function_dense MeasureTheory.Lp.boundedContinuousFunction_dense
 
 end Lp
 
@@ -380,13 +380,13 @@ variable (E) (μ)
 namespace BoundedContinuousFunction
 
 theorem toLp_denseRange [μ.WeaklyRegular] [FiniteMeasure μ] :
-    DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] lp E p μ) :=
+    DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   rw [denseRange_iff_closure_range]
   suffices (LinearMap.range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
     by exact congr_arg coe this
-  simp [range_to_Lp p μ, MeasureTheory.lp.boundedContinuousFunction_dense E hp]
+  simp [range_to_Lp p μ, MeasureTheory.Lp.boundedContinuousFunction_dense E hp]
 #align bounded_continuous_function.to_Lp_dense_range BoundedContinuousFunction.toLp_denseRange
 
 end BoundedContinuousFunction
@@ -394,13 +394,13 @@ end BoundedContinuousFunction
 namespace ContinuousMap
 
 theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [FiniteMeasure μ] :
-    DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] lp E p μ) :=
+    DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   rw [denseRange_iff_closure_range]
   suffices (LinearMap.range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
     by exact congr_arg coe this
-  simp [range_to_Lp p μ, MeasureTheory.lp.boundedContinuousFunction_dense E hp]
+  simp [range_to_Lp p μ, MeasureTheory.Lp.boundedContinuousFunction_dense E hp]
 #align continuous_map.to_Lp_dense_range ContinuousMap.toLp_denseRange
 
 end ContinuousMap

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

@@ -62,7 +62,7 @@ Vitali-Carathéodory theorem, in the file `measure_theory.vitali_caratheodory`.
 -/
 
 
-open ENNReal NNReal Topology BoundedContinuousFunction
+open scoped ENNReal NNReal Topology BoundedContinuousFunction
 
 open MeasureTheory TopologicalSpace ContinuousMap Set
 

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

@@ -158,7 +158,7 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and consists of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp _ _ _ _ hε
+  apply hf.induction_dense hp _ _ _ _ hε;
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
@@ -259,7 +259,7 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp _ _ _ _ hε
+  apply hf.induction_dense hp _ _ _ _ hε;
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit 13bf7613c96a9fd66a81b9020a82cad9a6ea1fcf
+! leanprover-community/mathlib commit e0736bb5b48bdadbca19dbd857e12bee38ccfbb8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -124,7 +124,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     simp only [hgv hx, Pi.zero_apply, zero_smul]
   have gc_mem : mem_ℒp (fun x => g x • c) p μ :=
     by
-    apply mem_ℒp.smul_of_top_left (mem_ℒp_top_const _)
+    refine' mem_ℒp.smul_of_top_left (mem_ℒp_top_const _) _
     refine' ⟨g.continuous.ae_strongly_measurable, _⟩
     have : snorm (v.indicator fun x => (1 : ℝ)) p μ < ⊤ :=
       by

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

@@ -268,7 +268,7 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
     let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩
     exact (f' + g').bounded_range
   -- ae strong measurability
-  · exact fun f ⟨_, h, _⟩ => h.AeStronglyMeasurable
+  · exact fun f ⟨_, h, _⟩ => h.AEStronglyMeasurable
   -- We are left with approximating characteristic functions.
   -- This follows from `exists_continuous_snorm_sub_le_of_closed`.
   intro c t ht htμ ε hε

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

@@ -379,7 +379,7 @@ variable (E) (μ)
 
 namespace BoundedContinuousFunction
 
-theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
+theorem toLp_denseRange [μ.WeaklyRegular] [FiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
@@ -393,7 +393,7 @@ end BoundedContinuousFunction
 
 namespace ContinuousMap
 
-theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
+theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [FiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

mathlib commit https://github.com/leanprover-community/mathlib/commit/49b7f94aab3a3bdca1f9f34c5d818afb253b3993

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit a8c97ed34c07fcfd7ebc6b83179b8f687275eba9
+! leanprover-community/mathlib commit 13bf7613c96a9fd66a81b9020a82cad9a6ea1fcf
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,7 +16,7 @@ import Mathbin.MeasureTheory.Integral.Bochner
 /-!
 # Approximation in Lᵖ by continuous functions
 
-This file proves that bounded continuous functions are dense in `Lp E p μ`, for `1 ≤ p < ∞`, if the
+This file proves that bounded continuous functions are dense in `Lp E p μ`, for `p < ∞`, if the
 domain `α` of the functions is a normal topological space and the measure `μ` is weakly regular.
 It also proves the same results for approximation by continuous functions with compact support
 when the space is locally compact and `μ` is regular.
@@ -146,9 +146,9 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
-continuous functions when `1 ≤ p < ∞`, version in terms of `snorm`. -/
+continuous functions when `p < ∞`, version in terms of `snorm`. -/
 theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [μ.regular] (hp : p ≠ ∞)
-    (h'p : 1 ≤ p) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α → E, HasCompactSupport g ∧ snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ :=
   by
   suffices H :
@@ -158,7 +158,7 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and consists of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp h'p _ _ _ _ hε
+  apply hf.induction_dense hp _ _ _ _ hε
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
@@ -169,45 +169,32 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   -- We are left with approximating characteristic functions.
   -- This follows from `exists_continuous_snorm_sub_le_of_closed`.
   intro c t ht htμ ε hε
-  have h'ε : ε / 2 ≠ 0 := by simpa using hε
+  rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩
   obtain ⟨η, ηpos, hη⟩ :
-    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε / 2
-  exact exists_snorm_indicator_le hp c h'ε
+    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ δ
+  exact exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
   obtain ⟨s, st, s_compact, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsCompact s ∧ μ (t \ s) < η
   exact ht.exists_is_compact_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
-  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ ε / 2 :=
+  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ δ :=
     by
     rw [← snorm_neg, neg_sub, ← indicator_diff st]
     exact hη _ μs.le
   obtain ⟨k, k_compact, sk, -⟩ : ∃ k : Set α, IsCompact k ∧ s ⊆ interior k ∧ k ⊆ univ
   exact exists_compact_between s_compact isOpen_univ (subset_univ _)
   rcases exists_continuous_snorm_sub_le_of_closed hp s_compact.is_closed isOpen_interior sk hsμ.ne c
-      h'ε with
+      δpos.ne' with
     ⟨f, f_cont, I2, f_bound, f_support, f_mem⟩
   have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
-    calc
-      snorm (f - t.indicator fun y => c) p μ =
-          snorm ((f - s.indicator fun y => c) + ((s.indicator fun y => c) - t.indicator fun y => c))
-            p μ :=
-        by simp only [sub_add_sub_cancel]
-      _ ≤
-          snorm (f - s.indicator fun y => c) p μ +
-            snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ :=
-        by
-        refine' snorm_add_le _ _ h'p
-        ·
-          exact
-            f_mem.ae_strongly_measurable.sub
-              (ae_strongly_measurable_const.indicator s_compact.measurable_set)
-        ·
-          exact
-            (ae_strongly_measurable_const.indicator s_compact.measurable_set).sub
-              (ae_strongly_measurable_const.indicator ht)
-      _ ≤ ε / 2 + ε / 2 := (add_le_add I2 I1)
-      _ = ε := ENNReal.add_halves _
-      
+    by
+    convert(hδ _ _
+          (f_mem.ae_strongly_measurable.sub
+            (ae_strongly_measurable_const.indicator s_compact.measurable_set))
+          ((ae_strongly_measurable_const.indicator s_compact.measurable_set).sub
+            (ae_strongly_measurable_const.indicator ht))
+          I2 I1).le
+    simp only [sub_add_sub_cancel]
   refine' ⟨f, I3, f_cont, f_mem, HasCompactSupport.intro k_compact fun x hx => _⟩
   rw [← Function.nmem_support]
   contrapose! hx
@@ -215,9 +202,9 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
 #align measure_theory.mem_ℒp.exists_has_compact_support_snorm_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le
 
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
-continuous functions when `1 ≤ p < ∞`, version in terms of `∫`. -/
+continuous functions when `0 < p < ∞`, version in terms of `∫`. -/
 theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpace α] [μ.regular]
-    {p : ℝ} (h'p : 1 ≤ p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
+    {p : ℝ} (hp : 0 < p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α → E,
       HasCompactSupport g ∧
         (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ :=
@@ -225,19 +212,15 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
     simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
-  have B : 1 ≤ ENNReal.ofReal p :=
-    by
-    convert ENNReal.ofReal_le_ofReal h'p
-    exact ennreal.of_real_one.symm
-  rcases hf.exists_has_compact_support_snorm_sub_le ENNReal.coe_ne_top B A with
+  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
+  rcases hf.exists_has_compact_support_snorm_sub_le ENNReal.coe_ne_top A with
     ⟨g, g_support, hg, g_cont, g_mem⟩
   change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
   refine' ⟨g, g_support, _, g_cont, g_mem⟩
-  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm (zero_lt_one.trans_le B).ne' ENNReal.coe_ne_top,
-    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal (zero_le_one.trans h'p),
-    Real.rpow_le_rpow_iff _ hε.le _] at hg
-  · exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
-  · exact inv_pos.2 (zero_lt_one.trans_le h'p)
+  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
+    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
+  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
@@ -248,7 +231,7 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpac
       HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
-  exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top le_rfl hε
+  exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_has_compact_support_lintegral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
@@ -259,14 +242,14 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
     hf⊢
-  simpa using hf.exists_has_compact_support_integral_rpow_sub_le le_rfl hε
+  simpa using hf.exists_has_compact_support_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
-/-- Any function in `ℒp` can be approximated by bounded continuous functions when `1 ≤ p < ∞`,
+/-- Any function in `ℒp` can be approximated by bounded continuous functions when `p < ∞`,
 version in terms of `snorm`. -/
-theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) (h'p : 1 ≤ p)
-    {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
+    (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ε ∧ Memℒp g p μ :=
   by
   suffices H :
@@ -276,7 +259,7 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp h'p _ _ _ _ hε
+  apply hf.induction_dense hp _ _ _ _ hε
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩
@@ -289,68 +272,51 @@ theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp :
   -- We are left with approximating characteristic functions.
   -- This follows from `exists_continuous_snorm_sub_le_of_closed`.
   intro c t ht htμ ε hε
-  have h'ε : ε / 2 ≠ 0 := by simpa using hε
+  rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩
   obtain ⟨η, ηpos, hη⟩ :
-    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε / 2
-  exact exists_snorm_indicator_le hp c h'ε
+    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ δ
+  exact exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
   obtain ⟨s, st, s_closed, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsClosed s ∧ μ (t \ s) < η
   exact ht.exists_is_closed_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
-  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ ε / 2 :=
+  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ δ :=
     by
     rw [← snorm_neg, neg_sub, ← indicator_diff st]
     exact hη _ μs.le
   rcases exists_continuous_snorm_sub_le_of_closed hp s_closed isOpen_univ (subset_univ _) hsμ.ne c
-      h'ε with
+      δpos.ne' with
     ⟨f, f_cont, I2, f_bound, -, f_mem⟩
   have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
-    calc
-      snorm (f - t.indicator fun y => c) p μ =
-          snorm ((f - s.indicator fun y => c) + ((s.indicator fun y => c) - t.indicator fun y => c))
-            p μ :=
-        by simp only [sub_add_sub_cancel]
-      _ ≤
-          snorm (f - s.indicator fun y => c) p μ +
-            snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ :=
-        by
-        refine' snorm_add_le _ _ h'p
-        ·
-          exact
-            f_mem.ae_strongly_measurable.sub
-              (ae_strongly_measurable_const.indicator s_closed.measurable_set)
-        ·
-          exact
-            (ae_strongly_measurable_const.indicator s_closed.measurable_set).sub
-              (ae_strongly_measurable_const.indicator ht)
-      _ ≤ ε / 2 + ε / 2 := (add_le_add I2 I1)
-      _ = ε := ENNReal.add_halves _
-      
+    by
+    convert(hδ _ _
+          (f_mem.ae_strongly_measurable.sub
+            (ae_strongly_measurable_const.indicator s_closed.measurable_set))
+          ((ae_strongly_measurable_const.indicator s_closed.measurable_set).sub
+            (ae_strongly_measurable_const.indicator ht))
+          I2 I1).le
+    simp only [sub_add_sub_cancel]
   refine' ⟨f, I3, f_cont, f_mem, _⟩
   exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
 #align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_snorm_sub_le
 
-/-- Any function in `ℒp` can be approximated by bounded continuous functions when `1 ≤ p < ∞`,
+/-- Any function in `ℒp` can be approximated by bounded continuous functions when `0 < p < ∞`,
 version in terms of `∫`. -/
-theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (h'p : 1 ≤ p)
+theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
     {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ :=
   by
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
     simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
-  have B : 1 ≤ ENNReal.ofReal p :=
-    by
-    convert ENNReal.ofReal_le_ofReal h'p
-    exact ennreal.of_real_one.symm
-  rcases hf.exists_bounded_continuous_snorm_sub_le ENNReal.coe_ne_top B A with ⟨g, hg, g_mem⟩
+  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
+  rcases hf.exists_bounded_continuous_snorm_sub_le ENNReal.coe_ne_top A with ⟨g, hg, g_mem⟩
   change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
   refine' ⟨g, _, g_mem⟩
-  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm (zero_lt_one.trans_le B).ne' ENNReal.coe_ne_top,
-    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal (zero_le_one.trans h'p),
-    Real.rpow_le_rpow_iff _ hε.le _] at hg
-  · exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
-  · exact inv_pos.2 (zero_lt_one.trans_le h'p)
+  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
+    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
+    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
+  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_integral_rpow_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,
@@ -360,7 +326,7 @@ theorem Integrable.exists_bounded_continuous_lintegral_sub_le [μ.WeaklyRegular]
     ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ :=
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
-  exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top le_rfl hε
+  exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top hε
 #align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_lintegral_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,
@@ -371,7 +337,7 @@ theorem Integrable.exists_bounded_continuous_integral_sub_le [μ.WeaklyRegular]
   by
   simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
     hf⊢
-  simpa using hf.exists_bounded_continuous_integral_rpow_sub_le le_rfl hε
+  simpa using hf.exists_bounded_continuous_integral_rpow_sub_le zero_lt_one hε
 #align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_integral_sub_le
 
 namespace Lp
@@ -389,7 +355,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   intro ε hε
   have A : ENNReal.ofReal ε ≠ 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, hε]
   obtain ⟨g, hg, g_mem⟩ : ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ENNReal.ofReal ε ∧ mem_ℒp g p μ
-  exact (Lp.mem_ℒp f).exists_bounded_continuous_snorm_sub_le hp _i.out A
+  exact (Lp.mem_ℒp f).exists_bounded_continuous_snorm_sub_le hp A
   refine' ⟨g_mem.to_Lp _, _, ⟨g, rfl⟩⟩
   simp only [dist_eq_norm, Metric.mem_closedBall']
   rw [Lp.norm_def]

mathlib commit https://github.com/leanprover-community/mathlib/commit/c9236f47f5b9df573443aa499c0d3968769628b7

@@ -4,21 +4,37 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
+! leanprover-community/mathlib commit a8c97ed34c07fcfd7ebc6b83179b8f687275eba9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Measure.Regular
 import Mathbin.MeasureTheory.Function.SimpleFuncDenseLp
 import Mathbin.Topology.UrysohnsLemma
+import Mathbin.MeasureTheory.Integral.Bochner
 
 /-!
 # Approximation in Lᵖ by continuous functions
 
 This file proves that bounded continuous functions are dense in `Lp E p μ`, for `1 ≤ p < ∞`, if the
 domain `α` of the functions is a normal topological space and the measure `μ` is weakly regular.
+It also proves the same results for approximation by continuous functions with compact support
+when the space is locally compact and `μ` is regular.
+
+The result is presented in several versions. First concrete versions giving an approximation
+up to `ε` in these various contexts, and then abstract versions stating that the topological
+closure of the relevant subgroups of `Lp` are the whole space.
+
+* `mem_ℒp.exists_has_compact_support_snorm_sub_le` states that, in a locally compact space,
+  an `ℒp` function can be approximated by continuous functions with compact support,
+  in the sense that `snorm (f - g) p μ` is small.
+* `mem_ℒp.exists_has_compact_support_integral_rpow_sub_le`: same result, but expressed in
+  terms of `∫ ‖f - g‖^p`.
+
+Versions with `integrable` instead of `mem_ℒp` are specialized to the case `p = 1`.
+Versions with `bounded_continuous` instead of `has_compact_support` drop the locally
+compact assumption and give only approximation by a bounded continuous function.
 
-The result is presented in several versions:
 * `measure_theory.Lp.bounded_continuous_function_dense`: The subgroup
   `measure_theory.Lp.bounded_continuous_function` of `Lp E p μ`, the additive subgroup of
   `Lp E p μ` consisting of equivalence classes containing a continuous representative, is dense in
@@ -48,152 +64,362 @@ Vitali-Carathéodory theorem, in the file `measure_theory.vitali_caratheodory`.
 
 open ENNReal NNReal Topology BoundedContinuousFunction
 
-open MeasureTheory TopologicalSpace ContinuousMap
+open MeasureTheory TopologicalSpace ContinuousMap Set
 
 variable {α : Type _} [MeasurableSpace α] [TopologicalSpace α] [NormalSpace α] [BorelSpace α]
 
-variable (E : Type _) [NormedAddCommGroup E] [SecondCountableTopologyEither α E]
+variable {E : Type _} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
 
-variable {p : ℝ≥0∞} [_i : Fact (1 ≤ p)] (hp : p ≠ ∞) (μ : Measure α)
-
-include _i hp
-
-namespace MeasureTheory.lp
+namespace MeasureTheory
 
 variable [NormedSpace ℝ E]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (u «expr ⊇ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
-/-- A function in `Lp` can be approximated in `Lp` by continuous functions. -/
-theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
-    (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=
+/-- A variant of Urysohn's lemma, `ℒ^p` version, for an outer regular measure `μ`:
+consider two sets `s ⊆ u` which are respectively closed and open with `μ s < ∞`, and a vector `c`.
+Then one may find a continuous function `f` equal to `c` on `s` and to `0` outside of `u`,
+bounded by `‖c‖` everywhere, and such that the `ℒ^p` norm of `f - s.indicator (λ y, c)` is
+arbitrarily small. Additionally, this function `f` belongs to `ℒ^p`. -/
+theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
+    (s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
+    (hε : ε ≠ 0) :
+    ∃ f : α → E,
+      Continuous f ∧
+        snorm (fun x => f x - s.indicator (fun y => c) x) p μ ≤ ε ∧
+          (∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ Memℒp f p μ :=
   by
-  have hp₀ : 0 < p := lt_of_lt_of_le zero_lt_one _i.elim
-  have hp₀' : 0 ≤ 1 / p.to_real := div_nonneg zero_le_one ENNReal.toReal_nonneg
-  have hp₀'' : 0 < p.to_real := by
-    simpa [← ENNReal.toReal_lt_toReal ENNReal.zero_ne_top hp] using hp₀
-  -- It suffices to prove that scalar multiples of the indicator function of a finite-measure
-  -- measurable set can be approximated by continuous functions
-  suffices
-    ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤),
-      (Lp.simple_func.indicator_const p hs hμs.Ne c : Lp E p μ) ∈
-        (BoundedContinuousFunction E p μ).topologicalClosure
+  obtain ⟨η, η_pos, hη⟩ :
+    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε
+  exact exists_snorm_indicator_le hp c hε
+  have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
+  obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α)(H : V ⊇ s), IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η
+  exact s_closed.measurable_set.exists_is_open_diff_lt hs ηpos.ne'
+  let v := u ∩ V
+  have hsv : s ⊆ v := subset_inter hsu sV
+  have hμv : μ v < ∞ := (measure_mono (inter_subset_right _ _)).trans_lt h'V
+  obtain ⟨g, hgv, hgs, hg_range⟩ :=
+    exists_continuous_zero_one_of_closed (u_open.inter V_open).isClosed_compl s_closed
+      (disjoint_compl_left_iff.2 hsv)
+  -- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
+  -- that this is pointwise bounded by the indicator of the set `v \ s`, which has small measure.
+  have g_norm : ∀ x, ‖g x‖ = g x := fun x => by rw [Real.norm_eq_abs, abs_of_nonneg (hg_range x).1]
+  have gc_bd0 : ∀ x, ‖g x • c‖ ≤ ‖c‖ := by
+    intro x
+    simp only [norm_smul, g_norm x]
+    apply mul_le_of_le_one_left (norm_nonneg _)
+    exact (hg_range x).2
+  have gc_bd : ∀ x, ‖g x • c - s.indicator (fun x => c) x‖ ≤ ‖(v \ s).indicator (fun x => c) x‖ :=
     by
-    rw [AddSubgroup.eq_top_iff']
-    refine' Lp.induction hp _ _ _ _
-    · exact this
-    · exact fun f g hf hg hfg' => AddSubgroup.add_mem _
-    · exact AddSubgroup.isClosed_topologicalClosure _
-  -- Let `s` be a finite-measure measurable set, let's approximate `c` times its indicator function
-  intro c s hs hsμ
-  refine' mem_closure_iff_frequently.mpr _
-  rw [metric.nhds_basis_closed_ball.frequently_iff]
-  intro ε hε
-  -- A little bit of pre-emptive work, to find `η : ℝ≥0` which will be a margin small enough for
-  -- our purposes
-  obtain ⟨η, hη_pos, hη_le⟩ : ∃ η, 0 < η ∧ (↑(‖bit0 ‖c‖‖₊ * (2 * η) ^ (1 / p.to_real)) : ℝ) ≤ ε :=
+    intro x
+    by_cases hv : x ∈ v
+    · rw [← Set.diff_union_of_subset hsv] at hv
+      cases' hv with hsv hs
+      ·
+        simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
+          Set.indicator_of_mem] using gc_bd0 x
+      · simp [hgs hs, hs]
+    · simp [hgv hv, (fun h => hv (hsv h) : x ∉ s)]
+  have gc_support : (Function.support fun x : α => g x • c) ⊆ v :=
     by
-    have : Filter.Tendsto (fun x : ℝ≥0 => ‖bit0 ‖c‖‖₊ * (2 * x) ^ (1 / p.to_real)) (𝓝 0) (𝓝 0) :=
-      by
-      have : Filter.Tendsto (fun x : ℝ≥0 => 2 * x) (𝓝 0) (𝓝 (2 * 0)) :=
-        filter.tendsto_id.const_mul 2
-      convert((NNReal.continuousAt_rpow_const (Or.inr hp₀')).Tendsto.comp this).const_mul _
-      simp [hp₀''.ne']
-    let ε' : ℝ≥0 := ⟨ε, hε.le⟩
-    have hε' : 0 < ε' := by exact_mod_cast hε
-    obtain ⟨δ, hδ, hδε'⟩ :=
-      nnreal.nhds_zero_basis.eventually_iff.mp (eventually_le_of_tendsto_lt hε' this)
-    obtain ⟨η, hη, hηδ⟩ := exists_between hδ
-    refine' ⟨η, hη, _⟩
-    exact_mod_cast hδε' hηδ
-  have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 hη_pos
-  -- Use the regularity of the measure to `η`-approximate `s` by an open superset and a closed
-  -- subset
-  obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _)(_ : u ⊇ s), IsOpen u ∧ μ u < μ s + ↑η :=
+    refine' Function.support_subset_iff'.2 fun x hx => _
+    simp only [hgv hx, Pi.zero_apply, zero_smul]
+  have gc_mem : mem_ℒp (fun x => g x • c) p μ :=
     by
-    refine' s.exists_is_open_lt_of_lt _ _
-    simpa using ENNReal.add_lt_add_left hsμ.ne hη_pos'
-  obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _)(_ : F ⊆ s), IsClosed F ∧ μ s < μ F + ↑η :=
-    hs.exists_is_closed_lt_add hsμ.ne hη_pos'.ne'
-  have : Disjoint (uᶜ) F := (Fs.trans su).disjoint_compl_left
-  have h_μ_sdiff : μ (u \ F) ≤ 2 * η :=
+    apply mem_ℒp.smul_of_top_left (mem_ℒp_top_const _)
+    refine' ⟨g.continuous.ae_strongly_measurable, _⟩
+    have : snorm (v.indicator fun x => (1 : ℝ)) p μ < ⊤ :=
+      by
+      refine' (snorm_indicator_const_le _ _).trans_lt _
+      simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne.def,
+        ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and_iff, or_false_iff, not_and, not_lt,
+        ENNReal.toReal_nonneg, imp_true_iff]
+    refine' (snorm_mono fun x => _).trans_lt this
+    by_cases hx : x ∈ v
+    ·
+      simp only [hx, abs_of_nonneg (hg_range x).1, (hg_range x).2, Real.norm_eq_abs,
+        indicator_of_mem, CstarRing.norm_one]
+    · simp only [hgv hx, Pi.zero_apply, Real.norm_eq_abs, abs_zero, abs_nonneg]
+  refine'
+    ⟨fun x => g x • c, g.continuous.smul continuous_const, (snorm_mono gc_bd).trans _, gc_bd0,
+      gc_support.trans (inter_subset_left _ _), gc_mem⟩
+  exact hη _ ((measure_mono (diff_subset_diff (inter_subset_right _ _) subset.rfl)).trans hV.le)
+#align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
+continuous functions when `1 ≤ p < ∞`, version in terms of `snorm`. -/
+theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [μ.regular] (hp : p ≠ ∞)
+    (h'p : 1 ≤ p) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ g : α → E, HasCompactSupport g ∧ snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ :=
+  by
+  suffices H :
+    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ mem_ℒp g p μ ∧ HasCompactSupport g
+  · rcases H with ⟨g, hg, g_cont, g_mem, g_support⟩
+    exact ⟨g, g_support, hg, g_cont, g_mem⟩
+  -- It suffices to check that the set of functions we consider approximates characteristic
+  -- functions, is stable under addition and consists of ae strongly measurable functions.
+  -- First check the latter easy facts.
+  apply hf.induction_dense hp h'p _ _ _ _ hε
+  rotate_left
+  -- stability under addition
+  · rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
+    exact ⟨f_cont.add g_cont, f_mem.add g_mem, hf.add hg⟩
+  -- ae strong measurability
+  · rintro f ⟨f_cont, f_mem, hf⟩
+    exact f_mem.ae_strongly_measurable
+  -- We are left with approximating characteristic functions.
+  -- This follows from `exists_continuous_snorm_sub_le_of_closed`.
+  intro c t ht htμ ε hε
+  have h'ε : ε / 2 ≠ 0 := by simpa using hε
+  obtain ⟨η, ηpos, hη⟩ :
+    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε / 2
+  exact exists_snorm_indicator_le hp c h'ε
+  have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
+  obtain ⟨s, st, s_compact, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsCompact s ∧ μ (t \ s) < η
+  exact ht.exists_is_compact_diff_lt htμ.ne hη_pos'.ne'
+  have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
+  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ ε / 2 :=
     by
-    have hFμ : μ F < ⊤ := (measure_mono Fs).trans_lt hsμ
-    refine' ENNReal.le_of_add_le_add_left hFμ.ne _
-    have : μ u < μ F + ↑η + ↑η := μu.trans (ENNReal.add_lt_add_right ENNReal.coe_ne_top μF)
-    convert this.le using 1
-    · rw [add_comm, ← measure_union, Set.diff_union_of_subset (Fs.trans su)]
-      exacts[disjoint_sdiff_self_left, F_closed.measurable_set]
-    have : (2 : ℝ≥0∞) * η = η + η := by simpa using add_mul (1 : ℝ≥0∞) 1 η
-    rw [this]
-    abel
-  -- Apply Urysohn's lemma to get a continuous approximation to the characteristic function of
-  -- the set `s`
-  obtain ⟨g, hgu, hgF, hg_range⟩ :=
-    exists_continuous_zero_one_of_closed u_open.is_closed_compl F_closed this
-  -- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
-  -- that this is pointwise bounded by the indicator of the set `u \ F`
-  have g_norm : ∀ x, ‖g x‖ = g x := fun x => by rw [Real.norm_eq_abs, abs_of_nonneg (hg_range x).1]
-  have gc_bd :
-    ∀ x, ‖g x • c - s.indicator (fun x => c) x‖ ≤ ‖(u \ F).indicator (fun x => bit0 ‖c‖) x‖ :=
+    rw [← snorm_neg, neg_sub, ← indicator_diff st]
+    exact hη _ μs.le
+  obtain ⟨k, k_compact, sk, -⟩ : ∃ k : Set α, IsCompact k ∧ s ⊆ interior k ∧ k ⊆ univ
+  exact exists_compact_between s_compact isOpen_univ (subset_univ _)
+  rcases exists_continuous_snorm_sub_le_of_closed hp s_compact.is_closed isOpen_interior sk hsμ.ne c
+      h'ε with
+    ⟨f, f_cont, I2, f_bound, f_support, f_mem⟩
+  have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
+    calc
+      snorm (f - t.indicator fun y => c) p μ =
+          snorm ((f - s.indicator fun y => c) + ((s.indicator fun y => c) - t.indicator fun y => c))
+            p μ :=
+        by simp only [sub_add_sub_cancel]
+      _ ≤
+          snorm (f - s.indicator fun y => c) p μ +
+            snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ :=
+        by
+        refine' snorm_add_le _ _ h'p
+        ·
+          exact
+            f_mem.ae_strongly_measurable.sub
+              (ae_strongly_measurable_const.indicator s_compact.measurable_set)
+        ·
+          exact
+            (ae_strongly_measurable_const.indicator s_compact.measurable_set).sub
+              (ae_strongly_measurable_const.indicator ht)
+      _ ≤ ε / 2 + ε / 2 := (add_le_add I2 I1)
+      _ = ε := ENNReal.add_halves _
+      
+  refine' ⟨f, I3, f_cont, f_mem, HasCompactSupport.intro k_compact fun x hx => _⟩
+  rw [← Function.nmem_support]
+  contrapose! hx
+  exact interior_subset (f_support hx)
+#align measure_theory.mem_ℒp.exists_has_compact_support_snorm_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le
+
+/-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
+continuous functions when `1 ≤ p < ∞`, version in terms of `∫`. -/
+theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpace α] [μ.regular]
+    {p : ℝ} (h'p : 1 ≤ p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
+    ∃ g : α → E,
+      HasCompactSupport g ∧
+        (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ :=
+  by
+  have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
+  have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
+    simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
+  have B : 1 ≤ ENNReal.ofReal p :=
     by
-    intro x
-    by_cases hu : x ∈ u
-    · rw [← Set.diff_union_of_subset (Fs.trans su)] at hu
-      cases' hu with hFu hF
-      · refine' (norm_sub_le _ _).trans _
-        refine' (add_le_add_left (norm_indicator_le_norm_self (fun x => c) x) _).trans _
-        have h₀ : g x * ‖c‖ + ‖c‖ ≤ 2 * ‖c‖ := by
-          nlinarith [(hg_range x).1, (hg_range x).2, norm_nonneg c]
-        have h₁ : (2 : ℝ) * ‖c‖ = bit0 ‖c‖ := by simpa using add_mul (1 : ℝ) 1 ‖c‖
-        simp [hFu, norm_smul, h₀, ← h₁, g_norm x]
-      · simp [hgF hF, Fs hF]
-    · have : x ∉ s := fun h => hu (su h)
-      simp [hgu hu, this]
-  -- The rest is basically just `ennreal`-arithmetic
-  have gc_snorm :
-    snorm ((fun x => g x • c) - s.indicator fun x => c) p μ ≤
-      (↑(‖bit0 ‖c‖‖₊ * (2 * η) ^ (1 / p.to_real)) : ℝ≥0∞) :=
+    convert ENNReal.ofReal_le_ofReal h'p
+    exact ennreal.of_real_one.symm
+  rcases hf.exists_has_compact_support_snorm_sub_le ENNReal.coe_ne_top B A with
+    ⟨g, g_support, hg, g_cont, g_mem⟩
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
+  refine' ⟨g, g_support, _, g_cont, g_mem⟩
+  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm (zero_lt_one.trans_le B).ne' ENNReal.coe_ne_top,
+    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal (zero_le_one.trans h'p),
+    Real.rpow_le_rpow_iff _ hε.le _] at hg
+  · exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  · exact inv_pos.2 (zero_lt_one.trans_le h'p)
+#align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
+
+/-- In a locally compact space, any integrable function can be approximated by compactly supported
+continuous functions, version in terms of `∫⁻`. -/
+theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpace α] [μ.regular]
+    {f : α → E} (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ g : α → E,
+      HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
+  by
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
+  exact hf.exists_has_compact_support_snorm_sub_le ENNReal.one_ne_top le_rfl hε
+#align measure_theory.integrable.exists_has_compact_support_lintegral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
+
+/-- In a locally compact space, any integrable function can be approximated by compactly supported
+continuous functions, version in terms of `∫`. -/
+theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace α] [μ.regular]
+    {f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
+    ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ :=
+  by
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
+    hf⊢
+  simpa using hf.exists_has_compact_support_integral_rpow_sub_le le_rfl hε
+#align measure_theory.integrable.exists_has_compact_support_integral_sub_le MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
+
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » t) -/
+/-- Any function in `ℒp` can be approximated by bounded continuous functions when `1 ≤ p < ∞`,
+version in terms of `snorm`. -/
+theorem Memℒp.exists_bounded_continuous_snorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) (h'p : 1 ≤ p)
+    {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ε ∧ Memℒp g p μ :=
+  by
+  suffices H :
+    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ mem_ℒp g p μ ∧ Metric.Bounded (range g)
+  · rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
+    exact ⟨⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩, hg, g_mem⟩
+  -- It suffices to check that the set of functions we consider approximates characteristic
+  -- functions, is stable under addition and made of ae strongly measurable functions.
+  -- First check the latter easy facts.
+  apply hf.induction_dense hp h'p _ _ _ _ hε
+  rotate_left
+  -- stability under addition
+  · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩
+    refine' ⟨f_cont.add g_cont, f_mem.add g_mem, _⟩
+    let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.bounded_range_iff.1 f_bd⟩
+    let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩
+    exact (f' + g').bounded_range
+  -- ae strong measurability
+  · exact fun f ⟨_, h, _⟩ => h.AeStronglyMeasurable
+  -- We are left with approximating characteristic functions.
+  -- This follows from `exists_continuous_snorm_sub_le_of_closed`.
+  intro c t ht htμ ε hε
+  have h'ε : ε / 2 ≠ 0 := by simpa using hε
+  obtain ⟨η, ηpos, hη⟩ :
+    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun x => c) p μ ≤ ε / 2
+  exact exists_snorm_indicator_le hp c h'ε
+  have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
+  obtain ⟨s, st, s_closed, μs⟩ : ∃ (s : _)(_ : s ⊆ t), IsClosed s ∧ μ (t \ s) < η
+  exact ht.exists_is_closed_diff_lt htμ.ne hη_pos'.ne'
+  have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
+  have I1 : snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ ≤ ε / 2 :=
     by
-    refine' (snorm_mono_ae (Filter.eventually_of_forall gc_bd)).trans _
-    rw [snorm_indicator_const (u_open.sdiff F_closed).MeasurableSet hp₀.ne' hp]
-    push_cast [← ENNReal.coe_rpow_of_nonneg _ hp₀']
-    exact ENNReal.mul_left_mono (ENNReal.monotone_rpow_of_nonneg hp₀' h_μ_sdiff)
-  have gc_cont : Continuous fun x => g x • c := g.continuous.smul continuous_const
-  have gc_mem_ℒp : mem_ℒp (fun x => g x • c) p μ :=
+    rw [← snorm_neg, neg_sub, ← indicator_diff st]
+    exact hη _ μs.le
+  rcases exists_continuous_snorm_sub_le_of_closed hp s_closed isOpen_univ (subset_univ _) hsμ.ne c
+      h'ε with
+    ⟨f, f_cont, I2, f_bound, -, f_mem⟩
+  have I3 : snorm (f - t.indicator fun y => c) p μ ≤ ε :=
+    calc
+      snorm (f - t.indicator fun y => c) p μ =
+          snorm ((f - s.indicator fun y => c) + ((s.indicator fun y => c) - t.indicator fun y => c))
+            p μ :=
+        by simp only [sub_add_sub_cancel]
+      _ ≤
+          snorm (f - s.indicator fun y => c) p μ +
+            snorm ((s.indicator fun y => c) - t.indicator fun y => c) p μ :=
+        by
+        refine' snorm_add_le _ _ h'p
+        ·
+          exact
+            f_mem.ae_strongly_measurable.sub
+              (ae_strongly_measurable_const.indicator s_closed.measurable_set)
+        ·
+          exact
+            (ae_strongly_measurable_const.indicator s_closed.measurable_set).sub
+              (ae_strongly_measurable_const.indicator ht)
+      _ ≤ ε / 2 + ε / 2 := (add_le_add I2 I1)
+      _ = ε := ENNReal.add_halves _
+      
+  refine' ⟨f, I3, f_cont, f_mem, _⟩
+  exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
+#align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_snorm_sub_le
+
+/-- Any function in `ℒp` can be approximated by bounded continuous functions when `1 ≤ p < ∞`,
+version in terms of `∫`. -/
+theorem Memℒp.exists_bounded_continuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (h'p : 1 ≤ p)
+    {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
+    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ :=
+  by
+  have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
+  have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
+    simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
+  have B : 1 ≤ ENNReal.ofReal p :=
     by
-    have : mem_ℒp ((fun x => g x • c) - s.indicator fun x => c) p μ :=
-      ⟨gc_cont.ae_strongly_measurable.sub
-          (strongly_measurable_const.indicator hs).AeStronglyMeasurable,
-        gc_snorm.trans_lt ENNReal.coe_lt_top⟩
-    simpa using this.add (mem_ℒp_indicator_const p hs c (Or.inr hsμ.ne))
-  refine' ⟨gc_mem_ℒp.to_Lp _, _, _⟩
-  · rw [mem_closedBall_iff_norm]
-    refine' le_trans _ hη_le
-    rw [simple_func.coe_indicator_const, indicator_const_Lp, ← mem_ℒp.to_Lp_sub, Lp.norm_to_Lp]
-    exact ENNReal.toReal_le_coe_of_le_coe gc_snorm
-  · rw [SetLike.mem_coe, mem_bounded_continuous_function_iff]
-    refine' ⟨BoundedContinuousFunction.ofNormedAddCommGroup _ gc_cont ‖c‖ _, rfl⟩
-    intro x
-    have h₀ : g x * ‖c‖ ≤ ‖c‖ := by nlinarith [(hg_range x).1, (hg_range x).2, norm_nonneg c]
-    simp [norm_smul, g_norm x, h₀]
+    convert ENNReal.ofReal_le_ofReal h'p
+    exact ennreal.of_real_one.symm
+  rcases hf.exists_bounded_continuous_snorm_sub_le ENNReal.coe_ne_top B A with ⟨g, hg, g_mem⟩
+  change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
+  refine' ⟨g, _, g_mem⟩
+  rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm (zero_lt_one.trans_le B).ne' ENNReal.coe_ne_top,
+    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal (zero_le_one.trans h'p),
+    Real.rpow_le_rpow_iff _ hε.le _] at hg
+  · exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  · exact inv_pos.2 (zero_lt_one.trans_le h'p)
+#align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_bounded_continuous_integral_rpow_sub_le
+
+/-- Any integrable function can be approximated by bounded continuous functions,
+version in terms of `∫⁻`. -/
+theorem Integrable.exists_bounded_continuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
+    (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+    ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ :=
+  by
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf⊢
+  exact hf.exists_bounded_continuous_snorm_sub_le ENNReal.one_ne_top le_rfl hε
+#align measure_theory.integrable.exists_bounded_continuous_lintegral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_lintegral_sub_le
+
+/-- Any integrable function can be approximated by bounded continuous functions,
+version in terms of `∫`. -/
+theorem Integrable.exists_bounded_continuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
+    (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
+    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ :=
+  by
+  simp only [← mem_ℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one] at
+    hf⊢
+  simpa using hf.exists_bounded_continuous_integral_rpow_sub_le le_rfl hε
+#align measure_theory.integrable.exists_bounded_continuous_integral_sub_le MeasureTheory.Integrable.exists_bounded_continuous_integral_sub_le
+
+namespace Lp
+
+variable (E)
+
+/-- A function in `Lp` can be approximated in `Lp` by continuous functions. -/
+theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)]
+    (hp : p ≠ ∞) [μ.WeaklyRegular] : (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=
+  by
+  rw [AddSubgroup.eq_top_iff']
+  intro f
+  refine' mem_closure_iff_frequently.mpr _
+  rw [metric.nhds_basis_closed_ball.frequently_iff]
+  intro ε hε
+  have A : ENNReal.ofReal ε ≠ 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, hε]
+  obtain ⟨g, hg, g_mem⟩ : ∃ g : α →ᵇ E, snorm (f - g) p μ ≤ ENNReal.ofReal ε ∧ mem_ℒp g p μ
+  exact (Lp.mem_ℒp f).exists_bounded_continuous_snorm_sub_le hp _i.out A
+  refine' ⟨g_mem.to_Lp _, _, ⟨g, rfl⟩⟩
+  simp only [dist_eq_norm, Metric.mem_closedBall']
+  rw [Lp.norm_def]
+  convert ENNReal.toReal_le_of_le_ofReal hε.le hg using 2
+  apply snorm_congr_ae
+  filter_upwards [coe_fn_sub f (g_mem.to_Lp g), g_mem.coe_fn_to_Lp]with x hx h'x
+  simp only [hx, Pi.sub_apply, sub_right_inj, h'x]
 #align measure_theory.Lp.bounded_continuous_function_dense MeasureTheory.lp.boundedContinuousFunction_dense
 
-end MeasureTheory.lp
+end Lp
+
+end MeasureTheory
+
+variable [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠ ∞)
 
 variable (𝕜 : Type _) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E]
 
-namespace BoundedContinuousFunction
+include _i hp
 
-open LinearMap (range)
+variable (E) (μ)
+
+namespace BoundedContinuousFunction
 
 theorem toLp_denseRange [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   rw [denseRange_iff_closure_range]
-  suffices (range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤ by
-    exact congr_arg coe this
+  suffices (LinearMap.range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
+    by exact congr_arg coe this
   simp [range_to_Lp p μ, MeasureTheory.lp.boundedContinuousFunction_dense E hp]
 #align bounded_continuous_function.to_Lp_dense_range BoundedContinuousFunction.toLp_denseRange
 
@@ -201,15 +427,13 @@ end BoundedContinuousFunction
 
 namespace ContinuousMap
 
-open LinearMap (range)
-
 theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange ⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] lp E p μ) :=
   by
   haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   rw [denseRange_iff_closure_range]
-  suffices (range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤ by
-    exact congr_arg coe this
+  suffices (LinearMap.range (to_Lp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
+    by exact congr_arg coe this
   simp [range_to_Lp p μ, MeasureTheory.lp.boundedContinuousFunction_dense E hp]
 #align continuous_map.to_Lp_dense_range ContinuousMap.toLp_denseRange
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -97,7 +97,7 @@ theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
       by
       have : Filter.Tendsto (fun x : ℝ≥0 => 2 * x) (𝓝 0) (𝓝 (2 * 0)) :=
         filter.tendsto_id.const_mul 2
-      convert ((NNReal.continuousAt_rpow_const (Or.inr hp₀')).Tendsto.comp this).const_mul _
+      convert((NNReal.continuousAt_rpow_const (Or.inr hp₀')).Tendsto.comp this).const_mul _
       simp [hp₀''.ne']
     let ε' : ℝ≥0 := ⟨ε, hε.le⟩
     have hε' : 0 < ε' := by exact_mod_cast hε

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

@@ -62,8 +62,8 @@ namespace MeasureTheory.lp
 
 variable [NormedSpace ℝ E]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (u «expr ⊇ » s) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: 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:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 /-- A function in `Lp` can be approximated in `Lp` by continuous functions. -/
 theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
     (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=

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: Heather Macbeth
 
 ! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! 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.
 -/
@@ -68,7 +68,7 @@ variable [NormedSpace ℝ E]
 theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
     (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=
   by
-  have hp₀ : 0 < p := lt_of_lt_of_le ENNReal.zero_lt_one _i.elim
+  have hp₀ : 0 < p := lt_of_lt_of_le zero_lt_one _i.elim
   have hp₀' : 0 ≤ 1 / p.to_real := div_nonneg zero_le_one ENNReal.toReal_nonneg
   have hp₀'' : 0 < p.to_real := by
     simpa [← ENNReal.toReal_lt_toReal ENNReal.zero_ne_top hp] using hp₀

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

@@ -46,7 +46,7 @@ Vitali-Carathéodory theorem, in the file `measure_theory.vitali_caratheodory`.
 -/
 
 
-open Ennreal NNReal Topology BoundedContinuousFunction
+open ENNReal NNReal Topology BoundedContinuousFunction
 
 open MeasureTheory TopologicalSpace ContinuousMap
 
@@ -68,10 +68,10 @@ variable [NormedSpace ℝ E]
 theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
     (boundedContinuousFunction E p μ).topologicalClosure = ⊤ :=
   by
-  have hp₀ : 0 < p := lt_of_lt_of_le Ennreal.zero_lt_one _i.elim
-  have hp₀' : 0 ≤ 1 / p.to_real := div_nonneg zero_le_one Ennreal.toReal_nonneg
+  have hp₀ : 0 < p := lt_of_lt_of_le ENNReal.zero_lt_one _i.elim
+  have hp₀' : 0 ≤ 1 / p.to_real := div_nonneg zero_le_one ENNReal.toReal_nonneg
   have hp₀'' : 0 < p.to_real := by
-    simpa [← Ennreal.toReal_lt_toReal Ennreal.zero_ne_top hp] using hp₀
+    simpa [← ENNReal.toReal_lt_toReal ENNReal.zero_ne_top hp] using hp₀
   -- It suffices to prove that scalar multiples of the indicator function of a finite-measure
   -- measurable set can be approximated by continuous functions
   suffices
@@ -106,21 +106,21 @@ theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
     obtain ⟨η, hη, hηδ⟩ := exists_between hδ
     refine' ⟨η, hη, _⟩
     exact_mod_cast hδε' hηδ
-  have hη_pos' : (0 : ℝ≥0∞) < η := Ennreal.coe_pos.2 hη_pos
+  have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 hη_pos
   -- Use the regularity of the measure to `η`-approximate `s` by an open superset and a closed
   -- subset
   obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _)(_ : u ⊇ s), IsOpen u ∧ μ u < μ s + ↑η :=
     by
     refine' s.exists_is_open_lt_of_lt _ _
-    simpa using Ennreal.add_lt_add_left hsμ.ne hη_pos'
+    simpa using ENNReal.add_lt_add_left hsμ.ne hη_pos'
   obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _)(_ : F ⊆ s), IsClosed F ∧ μ s < μ F + ↑η :=
     hs.exists_is_closed_lt_add hsμ.ne hη_pos'.ne'
   have : Disjoint (uᶜ) F := (Fs.trans su).disjoint_compl_left
   have h_μ_sdiff : μ (u \ F) ≤ 2 * η :=
     by
     have hFμ : μ F < ⊤ := (measure_mono Fs).trans_lt hsμ
-    refine' Ennreal.le_of_add_le_add_left hFμ.ne _
-    have : μ u < μ F + ↑η + ↑η := μu.trans (Ennreal.add_lt_add_right Ennreal.coe_ne_top μF)
+    refine' ENNReal.le_of_add_le_add_left hFμ.ne _
+    have : μ u < μ F + ↑η + ↑η := μu.trans (ENNReal.add_lt_add_right ENNReal.coe_ne_top μF)
     convert this.le using 1
     · rw [add_comm, ← measure_union, Set.diff_union_of_subset (Fs.trans su)]
       exacts[disjoint_sdiff_self_left, F_closed.measurable_set]
@@ -157,21 +157,21 @@ theorem boundedContinuousFunction_dense [μ.WeaklyRegular] :
     by
     refine' (snorm_mono_ae (Filter.eventually_of_forall gc_bd)).trans _
     rw [snorm_indicator_const (u_open.sdiff F_closed).MeasurableSet hp₀.ne' hp]
-    push_cast [← Ennreal.coe_rpow_of_nonneg _ hp₀']
-    exact Ennreal.mul_left_mono (Ennreal.monotone_rpow_of_nonneg hp₀' h_μ_sdiff)
+    push_cast [← ENNReal.coe_rpow_of_nonneg _ hp₀']
+    exact ENNReal.mul_left_mono (ENNReal.monotone_rpow_of_nonneg hp₀' h_μ_sdiff)
   have gc_cont : Continuous fun x => g x • c := g.continuous.smul continuous_const
   have gc_mem_ℒp : mem_ℒp (fun x => g x • c) p μ :=
     by
     have : mem_ℒp ((fun x => g x • c) - s.indicator fun x => c) p μ :=
       ⟨gc_cont.ae_strongly_measurable.sub
           (strongly_measurable_const.indicator hs).AeStronglyMeasurable,
-        gc_snorm.trans_lt Ennreal.coe_lt_top⟩
+        gc_snorm.trans_lt ENNReal.coe_lt_top⟩
     simpa using this.add (mem_ℒp_indicator_const p hs c (Or.inr hsμ.ne))
   refine' ⟨gc_mem_ℒp.to_Lp _, _, _⟩
   · rw [mem_closedBall_iff_norm]
     refine' le_trans _ hη_le
     rw [simple_func.coe_indicator_const, indicator_const_Lp, ← mem_ℒp.to_Lp_sub, Lp.norm_to_Lp]
-    exact Ennreal.toReal_le_coe_of_le_coe gc_snorm
+    exact ENNReal.toReal_le_coe_of_le_coe gc_snorm
   · rw [SetLike.mem_coe, mem_bounded_continuous_function_iff]
     refine' ⟨BoundedContinuousFunction.ofNormedAddCommGroup _ gc_cont ‖c‖ _, rfl⟩
     intro x

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

@@ -120,7 +120,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     refine' ⟨g.continuous.aestronglyMeasurable, _⟩
     have : snorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ := by
       refine' (snorm_indicator_const_le _ _).trans_lt _
-      simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne.def,
+      simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne,
         ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and_iff, or_false_iff, not_and, not_lt,
         ENNReal.toReal_nonneg, imp_true_iff]
     refine' (snorm_mono fun x => _).trans_lt this
@@ -198,8 +198,8 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
         (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ := by
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
-    simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
-  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
+    simp only [Ne, ENNReal.ofReal_eq_zero, not_le, I]
+  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_hasCompactSupport_snorm_sub_le ENNReal.coe_ne_top A with
     ⟨g, g_support, hg, g_cont, g_mem⟩
   change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
@@ -292,8 +292,8 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ := by
   have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
   have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
-    simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]
-  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp
+    simp only [Ne, ENNReal.ofReal_eq_zero, not_le, I]
+  have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp
   rcases hf.exists_boundedContinuous_snorm_sub_le ENNReal.coe_ne_top A with ⟨g, hg, g_mem⟩
   change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg
   refine' ⟨g, _, g_mem⟩
@@ -334,7 +334,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   refine' mem_closure_iff_frequently.mpr _
   rw [Metric.nhds_basis_closedBall.frequently_iff]
   intro ε hε
-  have A : ENNReal.ofReal ε ≠ 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, hε]
+  have A : ENNReal.ofReal ε ≠ 0 := by simp only [Ne, ENNReal.ofReal_eq_zero, not_le, hε]
   obtain ⟨g, hg, g_mem⟩ :
       ∃ g : α →ᵇ E, snorm ((f : α → E) - (g : α → E)) p μ ≤ ENNReal.ofReal ε ∧ Memℒp g p μ :=
     (Lp.memℒp f).exists_boundedContinuous_snorm_sub_le hp A

Inspired by #10538 add a positivity extension for Bochner integrals.

@@ -207,7 +207,7 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
-  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
+  positivity
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
@@ -300,7 +300,7 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
-  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
+  positivity
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -64,7 +64,6 @@ open scoped ENNReal NNReal Topology BoundedContinuousFunction
 open MeasureTheory TopologicalSpace ContinuousMap Set Bornology
 
 variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [T4Space α] [BorelSpace α]
-
 variable {E : Type*} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
 
 namespace MeasureTheory
@@ -358,9 +357,7 @@ end Lp
 end MeasureTheory
 
 variable [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠ ∞)
-
 variable (𝕜 : Type*) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E]
-
 variable (E) (μ)
 
 namespace BoundedContinuousFunction

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -342,7 +342,7 @@ theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [_i
   refine' ⟨g_mem.toLp _, _, ⟨g, rfl⟩⟩
   simp only [dist_eq_norm, Metric.mem_closedBall']
   rw [Lp.norm_def]
-  -- porting note: original proof started with:
+  -- Porting note: original proof started with:
   -- convert ENNReal.toReal_le_of_le_ofReal hε.le hg using 2
   -- the `convert` was completely borked and timed out
   have key : snorm ((f : α → E) - (g : α → E)) p μ = snorm (f - Memℒp.toLp (↑g) g_mem) p μ := by

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -84,11 +84,11 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
         snorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
           (∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ Memℒp f p μ := by
   obtain ⟨η, η_pos, hη⟩ :
-    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ ε
-  exact exists_snorm_indicator_le hp c hε
+      ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ ε :=
+    exists_snorm_indicator_le hp c hε
   have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
-  obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η
-  exact s_closed.measurableSet.exists_isOpen_diff_lt hs ηpos.ne'
+  obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η :=
+    s_closed.measurableSet.exists_isOpen_diff_lt hs ηpos.ne'
   let v := u ∩ V
   have hsv : s ⊆ v := subset_inter hsu sV
   have hμv : μ v < ∞ := (measure_mono (inter_subset_right _ _)).trans_lt h'V
@@ -160,11 +160,11 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [WeaklyLocallyCompactSpace
   intro c t ht htμ ε hε
   rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩
   obtain ⟨η, ηpos, hη⟩ :
-    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ
-  exact exists_snorm_indicator_le hp c δpos.ne'
+      ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ :=
+    exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
-  obtain ⟨s, st, s_compact, μs⟩ : ∃ s, s ⊆ t ∧ IsCompact s ∧ μ (t \ s) < η
-  exact ht.exists_isCompact_diff_lt htμ.ne hη_pos'.ne'
+  obtain ⟨s, st, s_compact, μs⟩ : ∃ s, s ⊆ t ∧ IsCompact s ∧ μ (t \ s) < η :=
+    ht.exists_isCompact_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
   have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ := by
     rw [← snorm_neg, neg_sub, ← indicator_diff st]
@@ -261,11 +261,11 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
   intro c t ht htμ ε hε
   rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩
   obtain ⟨η, ηpos, hη⟩ :
-    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ
-  exact exists_snorm_indicator_le hp c δpos.ne'
+      ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ :=
+    exists_snorm_indicator_le hp c δpos.ne'
   have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
-  obtain ⟨s, st, s_closed, μs⟩ : ∃ s, s ⊆ t ∧ IsClosed s ∧ μ (t \ s) < η
-  exact ht.exists_isClosed_diff_lt htμ.ne hη_pos'.ne'
+  obtain ⟨s, st, s_closed, μs⟩ : ∃ s, s ⊆ t ∧ IsClosed s ∧ μ (t \ s) < η :=
+    ht.exists_isClosed_diff_lt htμ.ne hη_pos'.ne'
   have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
   have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ := by
     rw [← snorm_neg, neg_sub, ← indicator_diff st]

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

This follows on from #6964.

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

@@ -141,8 +141,8 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [WeaklyLocallyCompactSpace
     (hp : p ≠ ∞) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α → E, HasCompactSupport g ∧ snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ := by
   suffices H :
-    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ HasCompactSupport g
-  · rcases H with ⟨g, hg, g_cont, g_mem, g_support⟩
+      ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ HasCompactSupport g by
+    rcases H with ⟨g, hg, g_cont, g_mem, g_support⟩
     exact ⟨g, g_support, hg, g_cont, g_mem⟩
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and consists of ae strongly measurable functions.
@@ -240,8 +240,8 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
     (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, snorm (f - (g : α → E)) p μ ≤ ε ∧ Memℒp g p μ := by
   suffices H :
-    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ IsBounded (range g)
-  · rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
+      ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ IsBounded (range g) by
+    rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
     exact ⟨⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩, hg, g_mem⟩
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.

This better matches other lemma names.

From LeanAPAP

@@ -208,7 +208,7 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
-  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_has_compact_support_integral_rpow_sub_le MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
@@ -301,7 +301,7 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
   rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,
     ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,
     Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg
-  exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
+  exact integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _
 #align measure_theory.mem_ℒp.exists_bounded_continuous_integral_rpow_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
 
 /-- Any integrable function can be approximated by bounded continuous functions,

Three particular examples which caught my eye; not exhaustive.

@@ -93,7 +93,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
   have hsv : s ⊆ v := subset_inter hsu sV
   have hμv : μ v < ∞ := (measure_mono (inter_subset_right _ _)).trans_lt h'V
   obtain ⟨g, hgv, hgs, hg_range⟩ :=
-    exists_continuous_zero_one_of_closed (u_open.inter V_open).isClosed_compl s_closed
+    exists_continuous_zero_one_of_isClosed (u_open.inter V_open).isClosed_compl s_closed
       (disjoint_compl_left_iff.2 hsv)
   -- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
   -- that this is pointwise bounded by the indicator of the set `v \ s`, which has small measure.

Use Bornology.IsBounded instead.

@@ -61,7 +61,7 @@ Vitali-Carathéodory theorem, in the file `Mathlib/MeasureTheory/Integral/Vitali
 
 open scoped ENNReal NNReal Topology BoundedContinuousFunction
 
-open MeasureTheory TopologicalSpace ContinuousMap Set
+open MeasureTheory TopologicalSpace ContinuousMap Set Bornology
 
 variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [T4Space α] [BorelSpace α]
 
@@ -240,9 +240,9 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
     (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α →ᵇ E, snorm (f - (g : α → E)) p μ ≤ ε ∧ Memℒp g p μ := by
   suffices H :
-    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ Metric.Bounded (range g)
+    ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ IsBounded (range g)
   · rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
-    exact ⟨⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩, hg, g_mem⟩
+    exact ⟨⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩, hg, g_mem⟩
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.
   -- First check the latter easy facts.
@@ -251,9 +251,9 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩
     refine' ⟨f_cont.add g_cont, f_mem.add g_mem, _⟩
-    let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.bounded_range_iff.1 f_bd⟩
-    let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩
-    exact (f' + g').bounded_range
+    let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.isBounded_range_iff.1 f_bd⟩
+    let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩
+    exact (f' + g').isBounded_range
   -- ae strong measurability
   · exact fun f ⟨_, h, _⟩ => h.aestronglyMeasurable
   -- We are left with approximating characteristic functions.
@@ -283,7 +283,7 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
           I2 I1).le using 2
     simp only [sub_add_sub_cancel]
   refine' ⟨f, I3, f_cont, f_mem, _⟩
-  exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range
+  exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).isBounded_range
 #align measure_theory.mem_ℒp.exists_bounded_continuous_snorm_sub_le MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le
 
 /-- Any function in `ℒp` can be approximated by bounded continuous functions when `0 < p < ∞`,

• Rename NormalSpace to T4Space.
• Add NormalSpace, a version without the T1Space assumption.
• Adjust some theorems.
• Supersedes thus closes #6892.
• Add some instance cycles, see #2030

@@ -63,7 +63,7 @@ open scoped ENNReal NNReal Topology BoundedContinuousFunction
 
 open MeasureTheory TopologicalSpace ContinuousMap Set
 
-variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [NormalSpace α] [BorelSpace α]
+variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [T4Space α] [BorelSpace α]
 
 variable {E : Type*} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
 

@@ -137,8 +137,8 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
 
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `p < ∞`, version in terms of `snorm`. -/
-theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [μ.Regular] (hp : p ≠ ∞)
-    {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
+theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [WeaklyLocallyCompactSpace α] [μ.Regular]
+    (hp : p ≠ ∞) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α → E, HasCompactSupport g ∧ snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ := by
   suffices H :
     ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ HasCompactSupport g
@@ -191,7 +191,8 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
 
 /-- In a locally compact space, any function in `ℒp` can be approximated by compactly supported
 continuous functions when `0 < p < ∞`, version in terms of `∫`. -/
-theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpace α] [μ.Regular]
+theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
+    [WeaklyLocallyCompactSpace α] [μ.Regular]
     {p : ℝ} (hp : 0 < p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α → E,
       HasCompactSupport g ∧
@@ -212,7 +213,8 @@ theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le [LocallyCompactSpa
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
 continuous functions, version in terms of `∫⁻`. -/
-theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpace α] [μ.Regular]
+theorem Integrable.exists_hasCompactSupport_lintegral_sub_le
+    [WeaklyLocallyCompactSpace α] [μ.Regular]
     {f : α → E} (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α → E,
       HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ := by
@@ -222,7 +224,8 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le [LocallyCompactSpac
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
 continuous functions, version in terms of `∫`. -/
-theorem Integrable.exists_hasCompactSupport_integral_sub_le [LocallyCompactSpace α] [μ.Regular]
+theorem Integrable.exists_hasCompactSupport_integral_sub_le
+    [WeaklyLocallyCompactSpace α] [μ.Regular]
     {f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧
       Continuous g ∧ Integrable g μ := by

@@ -169,8 +169,8 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ := by
     rw [← snorm_neg, neg_sub, ← indicator_diff st]
     exact hη _ μs.le
-  obtain ⟨k, k_compact, sk, -⟩ : ∃ k : Set α, IsCompact k ∧ s ⊆ interior k ∧ k ⊆ univ
-  exact exists_compact_between s_compact isOpen_univ (subset_univ _)
+  obtain ⟨k, k_compact, sk⟩ : ∃ k : Set α, IsCompact k ∧ s ⊆ interior k :=
+    exists_compact_superset s_compact
   rcases exists_continuous_snorm_sub_le_of_closed hp s_compact.isClosed isOpen_interior sk hsμ.ne c
       δpos.ne' with
     ⟨f, f_cont, I2, _f_bound, f_support, f_mem⟩

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

This has nice performance benefits.

@@ -63,9 +63,9 @@ open scoped ENNReal NNReal Topology BoundedContinuousFunction
 
 open MeasureTheory TopologicalSpace ContinuousMap Set
 
-variable {α : Type _} [MeasurableSpace α] [TopologicalSpace α] [NormalSpace α] [BorelSpace α]
+variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [NormalSpace α] [BorelSpace α]
 
-variable {E : Type _} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
+variable {E : Type*} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
 
 namespace MeasureTheory
 
@@ -356,7 +356,7 @@ end MeasureTheory
 
 variable [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (hp : p ≠ ∞)
 
-variable (𝕜 : Type _) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E]
+variable (𝕜 : Type*) [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 E]
 
 variable (E) (μ)
 

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module measure_theory.function.continuous_map_dense
-! leanprover-community/mathlib commit e0736bb5b48bdadbca19dbd857e12bee38ccfbb8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Measure.Regular
 import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
 import Mathlib.Topology.UrysohnsLemma
 import Mathlib.MeasureTheory.Integral.Bochner
 
+#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"e0736bb5b48bdadbca19dbd857e12bee38ccfbb8"
+
 /-!
 # Approximation in Lᵖ by continuous functions
 

@@ -379,13 +379,14 @@ end BoundedContinuousFunction
 
 namespace ContinuousMap
 
+/-- Continuous functions are dense in `MeasureTheory.Lp`, `1 ≤ p < ∞`. This theorem assumes that
+the domain is a compact space because otherwise `ContinuousMap.toLp` is undefined. Use
+`BoundedContinuousFunction.toLp_denseRange` if the domain is not a compact space.  -/
 theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
     DenseRange (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) := by
-  haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
-  rw [denseRange_iff_closure_range]
-  suffices (LinearMap.range (toLp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
-    by exact congr_arg ((↑) : AddSubgroup (Lp E p μ) → Set (Lp E p μ)) this
-  simpa [range_toLp p μ] using MeasureTheory.Lp.boundedContinuousFunction_dense E hp
+  refine (BoundedContinuousFunction.toLp_denseRange _ _ hp 𝕜).mono ?_
+  refine range_subset_iff.2 fun f ↦ ?_
+  exact ⟨f.toContinuousMap, rfl⟩
 set_option linter.uppercaseLean3 false in
 #align continuous_map.to_Lp_dense_range ContinuousMap.toLp_denseRange
 

@@ -25,13 +25,13 @@ The result is presented in several versions. First concrete versions giving an a
 up to `ε` in these various contexts, and then abstract versions stating that the topological
 closure of the relevant subgroups of `Lp` are the whole space.
 
-* `mem_ℒp.exists_has_compact_support_snorm_sub_le` states that, in a locally compact space,
-  an `ℒp` function can be approximated by continuous functions with compact support,
+* `MeasureTheory.Memℒp.exists_hasCompactSupport_snorm_sub_le` states that, in a locally compact
+  space, an `ℒp` function can be approximated by continuous functions with compact support,
   in the sense that `snorm (f - g) p μ` is small.
-* `mem_ℒp.exists_has_compact_support_integral_rpow_sub_le`: same result, but expressed in
+* `MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le`: same result, but expressed in
   terms of `∫ ‖f - g‖^p`.
 
-Versions with `integrable` instead of `mem_ℒp` are specialized to the case `p = 1`.
+Versions with `Integrable` instead of `Memℒp` are specialized to the case `p = 1`.
 Versions with `boundedContinuous` instead of `HasCompactSupport` drop the locally
 compact assumption and give only approximation by a bounded continuous function.
 
@@ -57,7 +57,7 @@ continuous function interpolating between these two sets.
 
 Are you looking for a result on "directional" approximation (above or below with respect to an
 order) of functions whose codomain is `ℝ≥0∞` or `ℝ`, by semicontinuous functions?  See the
-Vitali-Carathéodory theorem, in the file `measure_theory.vitali_caratheodory`.
+Vitali-Carathéodory theorem, in the file `Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean`.
 
 -/
 
@@ -112,8 +112,7 @@ theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ 
     by_cases hv : x ∈ v
     · rw [← Set.diff_union_of_subset hsv] at hv
       cases' hv with hsv hs
-      ·
-        simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
+      · simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
           Set.indicator_of_mem] using gc_bd0 x
       · simp [hgs hs, hs]
     · simp [hgv hv, show x ∉ s from fun h => hv (hsv h)]
@@ -151,7 +150,7 @@ theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [LocallyCompactSpace α] [
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and consists of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp _ _ _ _ hε;
+  apply hf.induction_dense hp _ _ _ _ hε
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
@@ -247,7 +246,7 @@ theorem Memℒp.exists_boundedContinuous_snorm_sub_le [μ.WeaklyRegular] (hp : p
   -- It suffices to check that the set of functions we consider approximates characteristic
   -- functions, is stable under addition and made of ae strongly measurable functions.
   -- First check the latter easy facts.
-  apply hf.induction_dense hp _ _ _ _ hε;
+  apply hf.induction_dense hp _ _ _ _ hε
   rotate_left
   -- stability under addition
   · rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩

Found with git grep -n "λ [a-zA-Z_ ]*,"

@@ -77,7 +77,7 @@ variable [NormedSpace ℝ E]
 /-- A variant of Urysohn's lemma, `ℒ^p` version, for an outer regular measure `μ`:
 consider two sets `s ⊆ u` which are respectively closed and open with `μ s < ∞`, and a vector `c`.
 Then one may find a continuous function `f` equal to `c` on `s` and to `0` outside of `u`,
-bounded by `‖c‖` everywhere, and such that the `ℒ^p` norm of `f - s.indicator (λ y, c)` is
+bounded by `‖c‖` everywhere, and such that the `ℒ^p` norm of `f - s.indicator (fun y ↦ c)` is
 arbitrarily small. Additionally, this function `f` belongs to `ℒ^p`. -/
 theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
     (s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}

Co-authored-by: Johan Commelin <johan@commelin.net>