measure_theory.integral.vitali_caratheodory ⟷ Mathlib.MeasureTheory.Integral.VitaliCaratheodory

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

@@ -6,7 +6,7 @@ Authors: Sébastien Gouëzel
 import MeasureTheory.Measure.Regular
 import Topology.Semicontinuous
 import MeasureTheory.Integral.Bochner
-import Topology.Instances.Ereal
+import Topology.Instances.EReal
 
 #align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
 
@@ -89,7 +89,7 @@ local infixr:25 " →ₛ " => SimpleFunc
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (u «expr ⊇ » s) -/
 #print MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
@@ -332,7 +332,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 #print MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of

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

@@ -250,9 +250,9 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
   refine' ⟨fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _⟩
   · by_cases h : x ∈ s
     · have := le_g1 x
-      simp only [h, Set.indicator_of_mem, top_le_iff] at this 
+      simp only [h, Set.indicator_of_mem, top_le_iff] at this
       simp [this]
-    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs ; exact hs h
+    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
       rw [this]
       exact (f_lt_g0 x).trans_le le_self_add
   ·
@@ -413,8 +413,8 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.biSup_add] at this  <;> [skip; exact ⟨0, fun x => by simp⟩]
-    simp only [lt_iSup_iff] at this 
+    erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => by simp⟩]
+    simp only [lt_iSup_iff] at this
     rcases this with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩
     convert int_fs.le
@@ -449,7 +449,7 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
           Integrable (fun x => (g x : ℝ)) μ ∧ ∫ x, (f x : ℝ) ∂μ - ε ≤ ∫ x, g x ∂μ :=
   by
   lift ε to ℝ≥0 using εpos.le
-  rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos 
+  rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
   have If : ∫⁻ x, f x ∂μ < ∞ := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral
   rcases exists_upper_semicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
   have Ig : ∫⁻ x, g x ∂μ < ∞ :=
@@ -586,8 +586,8 @@ theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → 
     ext x
     simp
   · simpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top
-  · simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint 
-    rw [add_comm] at gint 
+  · simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint
+    rw [add_comm] at gint
     simpa [integral_neg] using gint
 #align measure_theory.exists_upper_semicontinuous_lt_integral_gt MeasureTheory.exists_upperSemicontinuous_lt_integral_gt
 -/

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.MeasureTheory.Measure.Regular
-import Mathbin.Topology.Semicontinuous
-import Mathbin.MeasureTheory.Integral.Bochner
-import Mathbin.Topology.Instances.Ereal
+import MeasureTheory.Measure.Regular
+import Topology.Semicontinuous
+import MeasureTheory.Integral.Bochner
+import Topology.Instances.Ereal
 
 #align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
 
@@ -89,7 +89,7 @@ local infixr:25 " →ₛ " => SimpleFunc
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (u «expr ⊇ » s) -/
 #print MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
@@ -332,7 +332,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 #print MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module measure_theory.integral.vitali_caratheodory
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! 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.Topology.Semicontinuous
 import Mathbin.MeasureTheory.Integral.Bochner
 import Mathbin.Topology.Instances.Ereal
 
+#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
 /-!
 # Vitali-Carathéodory theorem
 
@@ -92,7 +89,7 @@ local infixr:25 " →ₛ " => SimpleFunc
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (u «expr ⊇ » s) -/
 #print MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
@@ -335,7 +332,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 #print MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of

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

@@ -87,13 +87,13 @@ variable {α : Type _} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α
 
 namespace MeasureTheory
 
--- mathport name: «expr →ₛ »
 local infixr:25 " →ₛ " => SimpleFunc
 
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+#print MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -158,9 +158,11 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
     conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
+-/
 
 open SimpleFunc (eapproxDiff tsum_eapproxDiff)
 
+#print MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge /-
 /-- Given a measurable function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -198,7 +200,9 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
         · intro n; exact (simple_func.measurable _).coe_nnreal_ennreal.AEMeasurable
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
+-/
 
+#print MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge /-
 /-- Given a measurable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
 lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `lintegral`.
@@ -227,7 +231,9 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
       _ ≤ ∫⁻ x : α, f x ∂μ + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
       _ = ∫⁻ x : α, f x ∂μ + ε := by rw [add_assoc, ENNReal.add_halves]
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
+-/
 
+#print MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable /-
 /-- Given an almost everywhere measurable function `f` with values in `ℝ≥0` in a sigma-finite space,
 there exists a lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `lintegral`.
@@ -266,9 +272,11 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
             lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
       _ = ∫⁻ x, f x ∂μ + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
+-/
 
 variable {μ}
 
+#print MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal /-
 /-- Given an integrable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
 lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `integral`.
@@ -322,11 +330,13 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
     · apply Filter.eventually_of_forall fun x => _; simp
     · apply gcont.measurable.ennreal_to_real.ae_measurable.ae_strongly_measurable
 #align measure_theory.exists_lt_lower_semicontinuous_integral_gt_nnreal MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal
+-/
 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+#print MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le /-
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -390,7 +400,9 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
     conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le
+-/
 
+#print MeasureTheory.exists_upperSemicontinuous_le_lintegral_le /-
 /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -425,7 +437,9 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     _ ≤ ∫⁻ x, g x ∂μ + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
     _ = ∫⁻ x, g x ∂μ + ε := by rw [add_assoc, ENNReal.add_halves]
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
+-/
 
+#print MeasureTheory.exists_upperSemicontinuous_le_integral_le /-
 /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `integral`.
@@ -460,10 +474,12 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
     · apply Filter.eventually_of_forall; simp
     · exact fint.ae_strongly_measurable
 #align measure_theory.exists_upper_semicontinuous_le_integral_le MeasureTheory.exists_upperSemicontinuous_le_integral_le
+-/
 
 /-! ### Vitali-Carathéodory theorem -/
 
 
+#print MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt /-
 /-- **Vitali-Carathéodory Theorem**: given an integrable real function `f`, there exists an
 integrable function `g > f` which is lower semicontinuous, with integral arbitrarily close
 to that of `f`. This function has to be `ereal`-valued in general. -/
@@ -547,7 +563,9 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
     · intro x
       exact EReal.continuousAt_add (by simp) (by simp)
 #align measure_theory.exists_lt_lower_semicontinuous_integral_lt MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt
+-/
 
+#print MeasureTheory.exists_upperSemicontinuous_lt_integral_gt /-
 /-- **Vitali-Carathéodory Theorem**: given an integrable real function `f`, there exists an
 integrable function `g < f` which is upper semicontinuous, with integral arbitrarily close to that
 of `f`. This function has to be `ereal`-valued in general. -/
@@ -575,6 +593,7 @@ theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → 
     rw [add_comm] at gint 
     simpa [integral_neg] using gint
 #align measure_theory.exists_upper_semicontinuous_lt_integral_gt MeasureTheory.exists_upperSemicontinuous_lt_integral_gt
+-/
 
 end MeasureTheory
 

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

@@ -100,11 +100,11 @@ function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation i
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
 theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
     (ε0 : ε ≠ 0) :
-    ∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
+    ∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ + ε :=
   by
   induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ H h₁ h₂ generalizing ε
   · let f := simple_func.piecewise s hs (simple_func.const α c) (simple_func.const α 0)
-    by_cases h : (∫⁻ x, f x ∂μ) = ⊤
+    by_cases h : ∫⁻ x, f x ∂μ = ⊤
     · refine'
         ⟨fun x => c, fun x => _, lowerSemicontinuous_const, by
           simp only [_root_.top_add, le_top, h]⟩
@@ -167,16 +167,14 @@ function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation i
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
 theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞}
     (εpos : ε ≠ 0) :
-    ∃ g : α → ℝ≥0∞,
-      (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
+    ∃ g : α → ℝ≥0∞, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ + ε :=
   by
   rcases ENNReal.exists_pos_sum_of_countable' εpos ℕ with ⟨δ, δpos, hδ⟩
   have :
     ∀ n,
       ∃ g : α → ℝ≥0,
         (∀ x, simple_func.eapprox_diff f n x ≤ g x) ∧
-          LowerSemicontinuous g ∧
-            (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, simple_func.eapprox_diff f n x ∂μ) + δ n :=
+          LowerSemicontinuous g ∧ ∫⁻ x, g x ∂μ ≤ ∫⁻ x, simple_func.eapprox_diff f n x ∂μ + δ n :=
     fun n =>
     simple_func.exists_le_lower_semicontinuous_lintegral_ge μ (simple_func.eapprox_diff f n)
       (δpos n).ne'
@@ -190,11 +188,11 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         ENNReal.coe_le_coe.2 hxy
   ·
     calc
-      (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
+      ∫⁻ x, ∑' n : ℕ, g n x ∂μ = ∑' n, ∫⁻ x, g n x ∂μ := by
         rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nnreal_ennreal.AEMeasurable]
-      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
-      _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
-      _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
+      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ + δ n) := (ENNReal.tsum_le_tsum hg)
+      _ = ∑' n, ∫⁻ x, eapprox_diff f n x ∂μ + ∑' n, δ n := ENNReal.tsum_add
+      _ ≤ ∫⁻ x : α, f x ∂μ + ε := by
         refine' add_le_add _ hδ.le
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
@@ -208,7 +206,7 @@ Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous
 theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α → ℝ≥0)
     (fmeas : Measurable f) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
     ∃ g : α → ℝ≥0∞,
-      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
+      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ + ε :=
   by
   have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
   rcases exists_pos_lintegral_lt_of_sigma_finite μ this with ⟨w, wpos, wmeas, wint⟩
@@ -223,11 +221,11 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
       _ ≤ g x := le_g x
   ·
     calc
-      (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
-      _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
+      ∫⁻ x : α, g x ∂μ ≤ ∫⁻ x : α, f x + w x ∂μ + ε / 2 := gint
+      _ = ∫⁻ x : α, f x ∂μ + ∫⁻ x : α, w x ∂μ + ε / 2 := by
         rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
-      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
-      _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
+      _ ≤ ∫⁻ x : α, f x ∂μ + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
+      _ = ∫⁻ x : α, f x ∂μ + ε := by rw [add_assoc, ENNReal.add_halves]
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
 
 /-- Given an almost everywhere measurable function `f` with values in `ℝ≥0` in a sigma-finite space,
@@ -237,7 +235,7 @@ Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous
 theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite μ] (f : α → ℝ≥0)
     (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
     ∃ g : α → ℝ≥0∞,
-      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
+      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ + ε :=
   by
   have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
   rcases exists_lt_lower_semicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
@@ -256,9 +254,9 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
       exact (f_lt_g0 x).trans_le le_self_add
   ·
     calc
-      (∫⁻ x, g0 x + g1 x ∂μ) = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
+      ∫⁻ x, g0 x + g1 x ∂μ = ∫⁻ x, g0 x ∂μ + ∫⁻ x, g1 x ∂μ :=
         lintegral_add_left g0_cont.measurable _
-      _ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) :=
+      _ ≤ ∫⁻ x, f x ∂μ + ε / 2 + (0 + ε / 2) :=
         by
         refine' add_le_add _ _
         · convert g0_int using 2
@@ -266,7 +264,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
         · convert g1_int
           simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
             lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
-      _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
+      _ = ∫⁻ x, f x ∂μ + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
 
 variable {μ}
@@ -281,7 +279,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
       (∀ x, (f x : ℝ≥0∞) < g x) ∧
         LowerSemicontinuous g ∧
           (∀ᵐ x ∂μ, g x < ⊤) ∧
-            Integrable (fun x => (g x).toReal) μ ∧ (∫ x, (g x).toReal ∂μ) < (∫ x, f x ∂μ) + ε :=
+            Integrable (fun x => (g x).toReal) μ ∧ ∫ x, (g x).toReal ∂μ < ∫ x, f x ∂μ + ε :=
   by
   have fmeas : AEMeasurable f μ :=
     by
@@ -289,14 +287,14 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
     simp only [Real.toNNReal_coe]
   lift ε to ℝ≥0 using εpos.le
   obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
-  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
+  have int_f_ne_top : ∫⁻ a : α, f a ∂μ ≠ ∞ :=
     (has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral).Ne
   rcases exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable μ f fmeas
       (ENNReal.coe_ne_zero.2 δpos.ne') with
     ⟨g, f_lt_g, gcont, gint⟩
-  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
+  have gint_ne : ∫⁻ x : α, g x ∂μ ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
   have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
-  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ :=
+  have Ig : ∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ a : α, g a ∂μ :=
     by
     apply lintegral_congr_ae
     filter_upwards [g_lt_top] with _ hx
@@ -311,7 +309,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
         ENNReal.toReal (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) =
             ENNReal.toReal (∫⁻ a : α, g a ∂μ) :=
           by congr 1
-        _ ≤ ENNReal.toReal ((∫⁻ a : α, f a ∂μ) + δ) :=
+        _ ≤ ENNReal.toReal (∫⁻ a : α, f a ∂μ + δ) :=
           by
           apply ENNReal.toReal_mono _ gint
           simpa using int_f_ne_top
@@ -334,8 +332,8 @@ function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation i
 `lintegral`.
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
 theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ≥0)
-    (int_f : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
-    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε :=
+    (int_f : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
+    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x ∂μ + ε :=
   by
   induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ H h₁ h₂ generalizing ε
   · let f := simple_func.piecewise s hs (simple_func.const α c) (simple_func.const α 0)
@@ -376,7 +374,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           simp_rw [mul_add]
           rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
           simpa using hc
-  · have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by
+  · have A : ∫⁻ x : α, f₁ x ∂μ + ∫⁻ x : α, f₂ x ∂μ ≠ ⊤ := by
       rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
     rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
       ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
@@ -397,12 +395,12 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
-theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : (∫⁻ x, f x ∂μ) ≠ ∞)
+theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : ∫⁻ x, f x ∂μ ≠ ∞)
     {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
-    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε :=
+    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x ∂μ + ε :=
   by
   obtain ⟨fs, fs_le_f, int_fs⟩ :
-    ∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 :=
+    ∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ ∫⁻ x, f x ∂μ ≤ ∫⁻ x, fs x ∂μ + ε / 2 :=
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
@@ -413,19 +411,19 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     convert int_fs.le
     rw [← simple_func.lintegral_eq_lintegral]
     rfl
-  have int_fs_lt_top : (∫⁻ x, fs x ∂μ) ≠ ∞ :=
+  have int_fs_lt_top : ∫⁻ x, fs x ∂μ ≠ ∞ :=
     by
     apply ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)
     simpa only [ENNReal.coe_le_coe] using fs_le_f x
   obtain ⟨g, g_le_fs, gcont, gint⟩ :
     ∃ g : α → ℝ≥0,
-      (∀ x, g x ≤ fs x) ∧ UpperSemicontinuous g ∧ (∫⁻ x, fs x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε / 2 :=
+      (∀ x, g x ≤ fs x) ∧ UpperSemicontinuous g ∧ ∫⁻ x, fs x ∂μ ≤ ∫⁻ x, g x ∂μ + ε / 2 :=
     fs.exists_upper_semicontinuous_le_lintegral_le int_fs_lt_top (ENNReal.half_pos ε0).ne'
   refine' ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _⟩
   calc
-    (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
-    _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
-    _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
+    ∫⁻ x, f x ∂μ ≤ ∫⁻ x, fs x ∂μ + ε / 2 := int_fs
+    _ ≤ ∫⁻ x, g x ∂μ + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
+    _ = ∫⁻ x, g x ∂μ + ε := by rw [add_assoc, ENNReal.add_halves]
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
 
 /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
@@ -437,13 +435,13 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
     ∃ g : α → ℝ≥0,
       (∀ x, g x ≤ f x) ∧
         UpperSemicontinuous g ∧
-          Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, g x ∂μ :=
+          Integrable (fun x => (g x : ℝ)) μ ∧ ∫ x, (f x : ℝ) ∂μ - ε ≤ ∫ x, g x ∂μ :=
   by
   lift ε to ℝ≥0 using εpos.le
   rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos 
-  have If : (∫⁻ x, f x ∂μ) < ∞ := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral
+  have If : ∫⁻ x, f x ∂μ < ∞ := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral
   rcases exists_upper_semicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
-  have Ig : (∫⁻ x, g x ∂μ) < ∞ :=
+  have Ig : ∫⁻ x, g x ∂μ < ∞ :=
     by
     apply lt_of_le_of_lt (lintegral_mono fun x => _) If
     simpa using gf x
@@ -475,7 +473,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
       (∀ x, (f x : EReal) < g x) ∧
         LowerSemicontinuous g ∧
           Integrable (fun x => EReal.toReal (g x)) μ ∧
-            (∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, EReal.toReal (g x) ∂μ) < (∫ x, f x ∂μ) + ε :=
+            (∀ᵐ x ∂μ, g x < ⊤) ∧ ∫ x, EReal.toReal (g x) ∂μ < ∫ x, f x ∂μ + ε :=
   by
   let δ : ℝ≥0 := ⟨ε / 2, (half_pos εpos).le⟩
   have δpos : 0 < δ := half_pos εpos
@@ -498,24 +496,24 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
     convert gp_integrable.sub gm_integrable
     ext x
     simp
-  show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε;
+  show ∫ x : α, (g x).toReal ∂μ < ∫ x : α, f x ∂μ + ε;
   exact
     calc
-      (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
+      ∫ x : α, (g x).toReal ∂μ = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
         integral_congr_ae ae_g
-      _ = (∫ x : α, EReal.toReal (gp x) ∂μ) - ∫ x : α, gm x ∂μ :=
+      _ = ∫ x : α, EReal.toReal (gp x) ∂μ - ∫ x : α, gm x ∂μ :=
         by
         simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal, coe_coe]
         exact integral_sub gp_integrable gm_integrable
-      _ < (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ∫ x : α, gm x ∂μ :=
+      _ < ∫ x : α, ↑(fp x) ∂μ + ↑δ - ∫ x : α, gm x ∂μ :=
         by
         apply sub_lt_sub_right
         convert gpint
         simp only [EReal.toReal_coe_ennreal]
-      _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := (sub_le_sub_left gmint _)
-      _ = (∫ x : α, f x ∂μ) + 2 * δ := by
+      _ ≤ ∫ x : α, ↑(fp x) ∂μ + ↑δ - (∫ x : α, fm x ∂μ - δ) := (sub_le_sub_left gmint _)
+      _ = ∫ x : α, f x ∂μ + 2 * δ := by
         simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm]; ring
-      _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]
+      _ = ∫ x : α, f x ∂μ + ε := by congr 1; field_simp [δ, mul_comm]
   show ∀ᵐ x : α ∂μ, g x < ⊤
   · filter_upwards [gp_lt_top] with _ hx
     simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (EReal.add_lt_top _ _).Ne, lt_top_iff_ne_top,
@@ -559,7 +557,7 @@ theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → 
       (∀ x, (g x : EReal) < f x) ∧
         UpperSemicontinuous g ∧
           Integrable (fun x => EReal.toReal (g x)) μ ∧
-            (∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, EReal.toReal (g x) ∂μ) + ε :=
+            (∀ᵐ x ∂μ, ⊥ < g x) ∧ ∫ x, f x ∂μ < ∫ x, EReal.toReal (g x) ∂μ + ε :=
   by
   rcases exists_lt_lower_semicontinuous_integral_lt (fun x => -f x) hf.neg εpos with
     ⟨g, g_lt_f, gcont, g_integrable, g_lt_top, gint⟩

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

@@ -146,7 +146,6 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
           simp_rw [mul_add]
           rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
           simpa using hc
-        
   · rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
     rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
     refine'
@@ -200,7 +199,6 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
         · intro n; exact (simple_func.measurable _).coe_nnreal_ennreal.AEMeasurable
-      
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
 
 /-- Given a measurable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
@@ -223,7 +221,6 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
     calc
       (f x : ℝ≥0∞) < f' x := by simpa [← ENNReal.coe_lt_coe] using add_lt_add_left (wpos x) (f x)
       _ ≤ g x := le_g x
-      
   ·
     calc
       (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
@@ -231,7 +228,6 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
         rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
       _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
-      
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
 
 /-- Given an almost everywhere measurable function `f` with values in `ℝ≥0` in a sigma-finite space,
@@ -271,7 +267,6 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
           simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
             lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
       _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
-      
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
 
 variable {μ}
@@ -324,7 +319,6 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
           rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]
         _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := (add_lt_add_left hδε _)
         _ = (∫⁻ a : α, ENNReal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
-        
     · apply Filter.eventually_of_forall fun x => _; simp
     · exact fmeas.coe_nnreal_real.ae_strongly_measurable
     · apply Filter.eventually_of_forall fun x => _; simp
@@ -382,7 +376,6 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           simp_rw [mul_add]
           rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
           simpa using hc
-        
   · have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by
       rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
     rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
@@ -433,7 +426,6 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
     _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
     _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
-    
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
 
 /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
@@ -524,7 +516,6 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
       _ = (∫ x : α, f x ∂μ) + 2 * δ := by
         simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm]; ring
       _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]
-      
   show ∀ᵐ x : α ∂μ, g x < ⊤
   · filter_upwards [gp_lt_top] with _ hx
     simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (EReal.add_lt_top _ _).Ne, lt_top_iff_ne_top,

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

@@ -93,7 +93,7 @@ local infixr:25 " →ₛ " => SimpleFunc
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (u «expr ⊇ » s) -/
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -334,7 +334,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (F «expr ⊆ » s) -/
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module measure_theory.integral.vitali_caratheodory
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Topology.Instances.Ereal
 /-!
 # Vitali-Carathéodory theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Vitali-Carathéodory theorem asserts the following. Consider an integrable function `f : α → ℝ` on
 a space with a regular measure. Then there exists a function `g : α → ereal` such that `f x < g x`
 everywhere, `g` is lower semicontinuous, and the integral of `g` is arbitrarily close to that of

mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725

@@ -235,7 +235,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
 there exists a lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `lintegral`.
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
-theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable [SigmaFinite μ] (f : α → ℝ≥0)
+theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite μ] (f : α → ℝ≥0)
     (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
     ∃ g : α → ℝ≥0∞,
       (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
@@ -269,7 +269,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable [SigmaFinite
             lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
       _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
       
-#align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable
+#align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
 
 variable {μ}
 
@@ -277,7 +277,7 @@ variable {μ}
 lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `integral`.
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
-theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : α → ℝ≥0)
+theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
     (fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
     ∃ g : α → ℝ≥0∞,
       (∀ x, (f x : ℝ≥0∞) < g x) ∧
@@ -326,7 +326,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
     · exact fmeas.coe_nnreal_real.ae_strongly_measurable
     · apply Filter.eventually_of_forall fun x => _; simp
     · apply gcont.measurable.ennreal_to_real.ae_measurable.ae_strongly_measurable
-#align measure_theory.exists_lt_lower_semicontinuous_integral_gt_nnreal MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nNReal
+#align measure_theory.exists_lt_lower_semicontinuous_integral_gt_nnreal MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal
 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 

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

@@ -301,7 +301,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
   have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ :=
     by
     apply lintegral_congr_ae
-    filter_upwards [g_lt_top]with _ hx
+    filter_upwards [g_lt_top] with _ hx
     simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
   refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
   · refine' ⟨gcont.measurable.ennreal_to_real.ae_measurable.ae_strongly_measurable, _⟩
@@ -495,7 +495,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   let g : α → EReal := fun x => (gp x : EReal) - gm x
   have ae_g : ∀ᵐ x ∂μ, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal :=
     by
-    filter_upwards [gp_lt_top]with _ hx
+    filter_upwards [gp_lt_top] with _ hx
     rw [EReal.toReal_sub] <;> simp [hx.ne]
   refine' ⟨g, _, _, _, _, _⟩
   show integrable (fun x => EReal.toReal (g x)) μ
@@ -523,7 +523,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
       _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]
       
   show ∀ᵐ x : α ∂μ, g x < ⊤
-  · filter_upwards [gp_lt_top]with _ hx
+  · filter_upwards [gp_lt_top] with _ hx
     simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (EReal.add_lt_top _ _).Ne, lt_top_iff_ne_top,
       lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,
       EReal.coe_ennreal_ne_bot]

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

@@ -125,7 +125,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
         or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
         simple_func.coe_zero, Set.piecewise_eq_indicator, simple_func.coe_piecewise, false_and_iff,
         restrict_apply] using h
-    obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _)(_ : u ⊇ s), IsOpen u ∧ μ u < μ s + ε / c :=
+    obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _) (_ : u ⊇ s), IsOpen u ∧ μ u < μ s + ε / c :=
       s.exists_is_open_lt_of_lt _ this
     refine'
       ⟨Set.indicator u fun x => c, fun x => _, u_open.lower_semicontinuous_indicator (zero_le _), _⟩
@@ -250,9 +250,9 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable [SigmaFinite
   refine' ⟨fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _⟩
   · by_cases h : x ∈ s
     · have := le_g1 x
-      simp only [h, Set.indicator_of_mem, top_le_iff] at this
+      simp only [h, Set.indicator_of_mem, top_le_iff] at this 
       simp [this]
-    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
+    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs ; exact hs h
       rw [this]
       exact (f_lt_g0 x).trans_le le_self_add
   ·
@@ -360,7 +360,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
         simple_func.coe_zero, Set.piecewise_eq_indicator, simple_func.coe_piecewise,
         false_and_iff] using int_f
     have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
-    obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _)(_ : F ⊆ s), IsClosed F ∧ μ s < μ F + ε / c :=
+    obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _) (_ : F ⊆ s), IsClosed F ∧ μ s < μ F + ε / c :=
       hs.exists_is_closed_lt_add μs_lt_top.ne this.ne'
     refine'
       ⟨Set.indicator F fun x => c, fun x => _, F_closed.upper_semicontinuous_indicator (zero_le _),
@@ -410,8 +410,8 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.biSup_add] at this <;> [skip;exact ⟨0, fun x => by simp⟩]
-    simp only [lt_iSup_iff] at this
+    erw [ENNReal.biSup_add] at this  <;> [skip; exact ⟨0, fun x => by simp⟩]
+    simp only [lt_iSup_iff] at this 
     rcases this with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩
     convert int_fs.le
@@ -445,7 +445,7 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
           Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, g x ∂μ :=
   by
   lift ε to ℝ≥0 using εpos.le
-  rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
+  rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos 
   have If : (∫⁻ x, f x ∂μ) < ∞ := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral
   rcases exists_upper_semicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
   have Ig : (∫⁻ x, g x ∂μ) < ∞ :=
@@ -579,8 +579,8 @@ theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → 
     ext x
     simp
   · simpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top
-  · simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint
-    rw [add_comm] at gint
+  · simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint 
+    rw [add_comm] at gint 
     simpa [integral_neg] using gint
 #align measure_theory.exists_upper_semicontinuous_lt_integral_gt MeasureTheory.exists_upperSemicontinuous_lt_integral_gt
 

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

@@ -75,7 +75,7 @@ See result `measure_theory.Lp.continuous_map_dense`, in the file
 -/
 
 
-open ENNReal NNReal
+open scoped ENNReal NNReal
 
 open MeasureTheory MeasureTheory.Measure
 

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

@@ -196,8 +196,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         refine' add_le_add _ hδ.le
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
-        · intro n
-          exact (simple_func.measurable _).coe_nnreal_ennreal.AEMeasurable
+        · intro n; exact (simple_func.measurable _).coe_nnreal_ennreal.AEMeasurable
       
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
 
@@ -253,9 +252,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable [SigmaFinite
     · have := le_g1 x
       simp only [h, Set.indicator_of_mem, top_le_iff] at this
       simp [this]
-    · have : f x = fmeas.mk f x := by
-        rw [Set.compl_subset_comm] at hs
-        exact hs h
+    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
       rw [this]
       exact (f_lt_g0 x).trans_le le_self_add
   ·
@@ -290,12 +287,10 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
   by
   have fmeas : AEMeasurable f μ :=
     by
-    convert fint.ae_strongly_measurable.real_to_nnreal.ae_measurable
-    ext1 x
+    convert fint.ae_strongly_measurable.real_to_nnreal.ae_measurable; ext1 x
     simp only [Real.toNNReal_coe]
   lift ε to ℝ≥0 using εpos.le
-  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε
-  exact exists_between εpos
+  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
   have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
     (has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral).Ne
   rcases exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable μ f fmeas
@@ -327,11 +322,9 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
         _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := (add_lt_add_left hδε _)
         _ = (∫⁻ a : α, ENNReal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
         
-    · apply Filter.eventually_of_forall fun x => _
-      simp
+    · apply Filter.eventually_of_forall fun x => _; simp
     · exact fmeas.coe_nnreal_real.ae_strongly_measurable
-    · apply Filter.eventually_of_forall fun x => _
-      simp
+    · apply Filter.eventually_of_forall fun x => _; simp
     · apply gcont.measurable.ennreal_to_real.ae_measurable.ae_strongly_measurable
 #align measure_theory.exists_lt_lower_semicontinuous_integral_gt_nnreal MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nNReal
 
@@ -467,14 +460,11 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
     · rw [sub_le_iff_le_add]
       convert ENNReal.toReal_mono _ gint
       · simp
-      · rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]
-        simp
+      · rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]; simp
       · simpa using Ig.ne
-    · apply Filter.eventually_of_forall
-      simp
+    · apply Filter.eventually_of_forall; simp
     · exact gcont.measurable.coe_nnreal_real.ae_measurable.ae_strongly_measurable
-    · apply Filter.eventually_of_forall
-      simp
+    · apply Filter.eventually_of_forall; simp
     · exact fint.ae_strongly_measurable
 #align measure_theory.exists_upper_semicontinuous_le_integral_le MeasureTheory.exists_upperSemicontinuous_le_integral_le
 
@@ -513,7 +503,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
     convert gp_integrable.sub gm_integrable
     ext x
     simp
-  show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε
+  show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε;
   exact
     calc
       (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
@@ -528,13 +518,9 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
         convert gpint
         simp only [EReal.toReal_coe_ennreal]
       _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := (sub_le_sub_left gmint _)
-      _ = (∫ x : α, f x ∂μ) + 2 * δ :=
-        by
-        simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm]
-        ring
-      _ = (∫ x : α, f x ∂μ) + ε := by
-        congr 1
-        field_simp [δ, mul_comm]
+      _ = (∫ x : α, f x ∂μ) + 2 * δ := by
+        simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm]; ring
+      _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]
       
   show ∀ᵐ x : α ∂μ, g x < ⊤
   · filter_upwards [gp_lt_top]with _ hx
@@ -545,8 +531,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   · intro x
     rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]
     refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _
-    · simp only [EReal.coe_ennreal_lt_coe_ennreal_iff, coe_coe]
-      exact fp_lt_gp x
+    · simp only [EReal.coe_ennreal_lt_coe_ennreal_iff, coe_coe]; exact fp_lt_gp x
     · simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff, coe_coe]
       exact gm_le_fm x
     · simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff, coe_coe]

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

@@ -417,7 +417,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.biSup_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
+    erw [ENNReal.biSup_add] at this <;> [skip;exact ⟨0, fun x => by simp⟩]
     simp only [lt_iSup_iff] at this
     rcases this with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩

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

@@ -189,7 +189,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
   ·
     calc
       (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
-        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_eNNReal.AEMeasurable]
+        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nnreal_ennreal.AEMeasurable]
       _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
       _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
@@ -197,7 +197,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
         · intro n
-          exact (simple_func.measurable _).coe_nNReal_eNNReal.AEMeasurable
+          exact (simple_func.measurable _).coe_nnreal_ennreal.AEMeasurable
       
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
 
@@ -213,7 +213,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
   have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
   rcases exists_pos_lintegral_lt_of_sigma_finite μ this with ⟨w, wpos, wmeas, wint⟩
   let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
-  rcases exists_le_lower_semicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nNReal_eNNReal
+  rcases exists_le_lower_semicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
       this with
     ⟨g, le_g, gcont, gint⟩
   refine' ⟨g, fun x => _, gcont, _⟩

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

@@ -417,8 +417,8 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.bsupᵢ_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
-    simp only [lt_supᵢ_iff] at this
+    erw [ENNReal.biSup_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
+    simp only [lt_iSup_iff] at this
     rcases this with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩
     convert int_fs.le

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

@@ -189,7 +189,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
   ·
     calc
       (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
-        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_eNNReal.AeMeasurable]
+        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_eNNReal.AEMeasurable]
       _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
       _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
@@ -197,7 +197,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
         · intro n
-          exact (simple_func.measurable _).coe_nNReal_eNNReal.AeMeasurable
+          exact (simple_func.measurable _).coe_nNReal_eNNReal.AEMeasurable
       
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
 
@@ -236,8 +236,8 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
 there exists a lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
 Formulation in terms of `lintegral`.
 Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
-theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable [SigmaFinite μ] (f : α → ℝ≥0)
-    (fmeas : AeMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
+theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable [SigmaFinite μ] (f : α → ℝ≥0)
+    (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
     ∃ g : α → ℝ≥0∞,
       (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
   by
@@ -272,7 +272,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable [SigmaFinite
             lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
       _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
       
-#align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable
+#align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aEMeasurable
 
 variable {μ}
 
@@ -288,7 +288,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
           (∀ᵐ x ∂μ, g x < ⊤) ∧
             Integrable (fun x => (g x).toReal) μ ∧ (∫ x, (g x).toReal ∂μ) < (∫ x, f x ∂μ) + ε :=
   by
-  have fmeas : AeMeasurable f μ :=
+  have fmeas : AEMeasurable f μ :=
     by
     convert fint.ae_strongly_measurable.real_to_nnreal.ae_measurable
     ext1 x

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -114,7 +114,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
         simp only [hc, Set.indicator_zero', Pi.zero_apply, simple_func.const_zero, imp_true_iff,
           eq_self_iff_true, simple_func.coe_zero, Set.piecewise_eq_indicator,
           simple_func.coe_piecewise, le_zero_iff]
-      · simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
+      · simp only [lintegral_const, MulZeroClass.zero_mul, zero_le, ENNReal.coe_zero]
     have : μ s < μ s + ε / c :=
       by
       have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
@@ -269,7 +269,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable [SigmaFinite
           exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
         · convert g1_int
           simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
-            lintegral_indicator, mul_zero, restrict_apply]
+            lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
       _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
       
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable
@@ -356,7 +356,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           eq_self_iff_true, simple_func.coe_zero, Set.piecewise_eq_indicator,
           simple_func.coe_piecewise, le_zero_iff]
       ·
-        simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,
+        simp only [hc, Set.indicator_zero', lintegral_const, MulZeroClass.zero_mul, Pi.zero_apply,
           simple_func.const_zero, zero_add, zero_le', simple_func.coe_zero,
           Set.piecewise_eq_indicator, ENNReal.coe_zero, simple_func.coe_piecewise, zero_le]
     have μs_lt_top : μ s < ∞ := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba

@@ -417,7 +417,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     by
     have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.bsupr_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
+    erw [ENNReal.bsupᵢ_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
     simp only [lt_supᵢ_iff] at this
     rcases this with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩

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

@@ -90,7 +90,7 @@ local infixr:25 " →ₛ " => SimpleFunc
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
 
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (u «expr ⊇ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (u «expr ⊇ » s) -/
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
 function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -190,7 +190,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
     calc
       (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
         rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_eNNReal.AeMeasurable]
-      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := ENNReal.tsum_le_tsum hg
+      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
       _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
         refine' add_le_add _ hδ.le
@@ -227,7 +227,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
       (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
       _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
         rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
-      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := add_le_add_right (add_le_add_left wint.le _) _
+      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
       _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
       
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
@@ -324,7 +324,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
           simpa using int_f_ne_top
         _ = ENNReal.toReal (∫⁻ a : α, f a ∂μ) + δ := by
           rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]
-        _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := add_lt_add_left hδε _
+        _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := (add_lt_add_left hδε _)
         _ = (∫⁻ a : α, ENNReal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
         
     · apply Filter.eventually_of_forall fun x => _
@@ -338,7 +338,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
 /-! ### Upper semicontinuous lower bound for nonnegative functions -/
 
 
-/- ./././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 (F «expr ⊆ » s) -/
 /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
 function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
 `lintegral`.
@@ -435,7 +435,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
   refine' ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _⟩
   calc
     (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
-    _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := add_le_add gint le_rfl
+    _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
     _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
     
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
@@ -527,7 +527,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
         apply sub_lt_sub_right
         convert gpint
         simp only [EReal.toReal_coe_ennreal]
-      _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := sub_le_sub_left gmint _
+      _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := (sub_le_sub_left gmint _)
       _ = (∫ x : α, f x ∂μ) + 2 * δ :=
         by
         simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm]

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

@@ -486,11 +486,11 @@ integrable function `g > f` which is lower semicontinuous, with integral arbitra
 to that of `f`. This function has to be `ereal`-valued in general. -/
 theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
     {ε : ℝ} (εpos : 0 < ε) :
-    ∃ g : α → Ereal,
-      (∀ x, (f x : Ereal) < g x) ∧
+    ∃ g : α → EReal,
+      (∀ x, (f x : EReal) < g x) ∧
         LowerSemicontinuous g ∧
-          Integrable (fun x => Ereal.toReal (g x)) μ ∧
-            (∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, Ereal.toReal (g x) ∂μ) < (∫ x, f x ∂μ) + ε :=
+          Integrable (fun x => EReal.toReal (g x)) μ ∧
+            (∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, EReal.toReal (g x) ∂μ) < (∫ x, f x ∂μ) + ε :=
   by
   let δ : ℝ≥0 := ⟨ε / 2, (half_pos εpos).le⟩
   have δpos : 0 < δ := half_pos εpos
@@ -502,13 +502,13 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   have int_fm : integrable (fun x => (fm x : ℝ)) μ := hf.neg.real_to_nnreal
   rcases exists_upper_semicontinuous_le_integral_le fm int_fm δpos with
     ⟨gm, gm_le_fm, gmcont, gm_integrable, gmint⟩
-  let g : α → Ereal := fun x => (gp x : Ereal) - gm x
-  have ae_g : ∀ᵐ x ∂μ, (g x).toReal = (gp x : Ereal).toReal - (gm x : Ereal).toReal :=
+  let g : α → EReal := fun x => (gp x : EReal) - gm x
+  have ae_g : ∀ᵐ x ∂μ, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal :=
     by
     filter_upwards [gp_lt_top]with _ hx
-    rw [Ereal.toReal_sub] <;> simp [hx.ne]
+    rw [EReal.toReal_sub] <;> simp [hx.ne]
   refine' ⟨g, _, _, _, _, _⟩
-  show integrable (fun x => Ereal.toReal (g x)) μ
+  show integrable (fun x => EReal.toReal (g x)) μ
   · rw [integrable_congr ae_g]
     convert gp_integrable.sub gm_integrable
     ext x
@@ -516,17 +516,17 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε
   exact
     calc
-      (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, Ereal.toReal (gp x) - Ereal.toReal (gm x) ∂μ :=
+      (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
         integral_congr_ae ae_g
-      _ = (∫ x : α, Ereal.toReal (gp x) ∂μ) - ∫ x : α, gm x ∂μ :=
+      _ = (∫ x : α, EReal.toReal (gp x) ∂μ) - ∫ x : α, gm x ∂μ :=
         by
-        simp only [Ereal.toReal_coe_eNNReal, ENNReal.coe_toReal, coe_coe]
+        simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal, coe_coe]
         exact integral_sub gp_integrable gm_integrable
       _ < (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ∫ x : α, gm x ∂μ :=
         by
         apply sub_lt_sub_right
         convert gpint
-        simp only [Ereal.toReal_coe_eNNReal]
+        simp only [EReal.toReal_coe_ennreal]
       _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := sub_le_sub_left gmint _
       _ = (∫ x : α, f x ∂μ) + 2 * δ :=
         by
@@ -538,37 +538,37 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
       
   show ∀ᵐ x : α ∂μ, g x < ⊤
   · filter_upwards [gp_lt_top]with _ hx
-    simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (Ereal.add_lt_top _ _).Ne, lt_top_iff_ne_top,
-      lt_top_iff_ne_top.1 hx, Ereal.coe_eNNReal_eq_top_iff, not_false_iff, Ereal.neg_eq_top_iff,
-      Ereal.coe_eNNReal_ne_bot]
-  show ∀ x, (f x : Ereal) < g x
+    simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (EReal.add_lt_top _ _).Ne, lt_top_iff_ne_top,
+      lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,
+      EReal.coe_ennreal_ne_bot]
+  show ∀ x, (f x : EReal) < g x
   · intro x
-    rw [Ereal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]
-    refine' Ereal.sub_lt_sub_of_lt_of_le _ _ _ _
-    · simp only [Ereal.coe_eNNReal_lt_coe_eNNReal_iff, coe_coe]
+    rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]
+    refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _
+    · simp only [EReal.coe_ennreal_lt_coe_ennreal_iff, coe_coe]
       exact fp_lt_gp x
-    · simp only [ENNReal.coe_le_coe, Ereal.coe_eNNReal_le_coe_eNNReal_iff, coe_coe]
+    · simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff, coe_coe]
       exact gm_le_fm x
-    · simp only [Ereal.coe_eNNReal_ne_bot, Ne.def, not_false_iff, coe_coe]
-    · simp only [Ereal.coe_nNReal_ne_top, Ne.def, not_false_iff, coe_coe]
+    · simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff, coe_coe]
+    · simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff, coe_coe]
   show LowerSemicontinuous g
   · apply LowerSemicontinuous.add'
     ·
       exact
         continuous_coe_ennreal_ereal.comp_lower_semicontinuous gpcont fun x y hxy =>
-          Ereal.coe_eNNReal_le_coe_eNNReal_iff.2 hxy
+          EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy
     · apply
         ereal.continuous_neg.comp_upper_semicontinuous_antitone _ fun x y hxy =>
-          Ereal.neg_le_neg_iff.2 hxy
+          EReal.neg_le_neg_iff.2 hxy
       dsimp
       apply
         continuous_coe_ennreal_ereal.comp_upper_semicontinuous _ fun x y hxy =>
-          Ereal.coe_eNNReal_le_coe_eNNReal_iff.2 hxy
+          EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy
       exact
         ennreal.continuous_coe.comp_upper_semicontinuous gmcont fun x y hxy =>
           ENNReal.coe_le_coe.2 hxy
     · intro x
-      exact Ereal.continuousAt_add (by simp) (by simp)
+      exact EReal.continuousAt_add (by simp) (by simp)
 #align measure_theory.exists_lt_lower_semicontinuous_integral_lt MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt
 
 /-- **Vitali-Carathéodory Theorem**: given an integrable real function `f`, there exists an
@@ -576,20 +576,20 @@ integrable function `g < f` which is upper semicontinuous, with integral arbitra
 of `f`. This function has to be `ereal`-valued in general. -/
 theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
     {ε : ℝ} (εpos : 0 < ε) :
-    ∃ g : α → Ereal,
-      (∀ x, (g x : Ereal) < f x) ∧
+    ∃ g : α → EReal,
+      (∀ x, (g x : EReal) < f x) ∧
         UpperSemicontinuous g ∧
-          Integrable (fun x => Ereal.toReal (g x)) μ ∧
-            (∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, Ereal.toReal (g x) ∂μ) + ε :=
+          Integrable (fun x => EReal.toReal (g x)) μ ∧
+            (∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, EReal.toReal (g x) ∂μ) + ε :=
   by
   rcases exists_lt_lower_semicontinuous_integral_lt (fun x => -f x) hf.neg εpos with
     ⟨g, g_lt_f, gcont, g_integrable, g_lt_top, gint⟩
   refine' ⟨fun x => -g x, _, _, _, _, _⟩
-  · exact fun x => Ereal.neg_lt_iff_neg_lt.1 (by simpa only [Ereal.coe_neg] using g_lt_f x)
+  · exact fun x => EReal.neg_lt_iff_neg_lt.1 (by simpa only [EReal.coe_neg] using g_lt_f x)
   ·
     exact
       ereal.continuous_neg.comp_lower_semicontinuous_antitone gcont fun x y hxy =>
-        Ereal.neg_le_neg_iff.2 hxy
+        EReal.neg_le_neg_iff.2 hxy
   · convert g_integrable.neg
     ext x
     simp

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: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module measure_theory.integral.vitali_caratheodory
-! leanprover-community/mathlib commit 83871f6cff322a1eafaa382ef70e9515fe240abd
+! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -104,7 +104,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
     by_cases h : (∫⁻ x, f x ∂μ) = ⊤
     · refine'
         ⟨fun x => c, fun x => _, lowerSemicontinuous_const, by
-          simp only [ENNReal.top_add, le_top, h]⟩
+          simp only [_root_.top_add, le_top, h]⟩
       simp only [simple_func.coe_const, simple_func.const_zero, simple_func.coe_zero,
         Set.piecewise_eq_indicator, simple_func.coe_piecewise]
       exact Set.indicator_le_self _ _ _
@@ -138,7 +138,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
           simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,
           Set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply]
       calc
-        (c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := ENNReal.mul_le_mul le_rfl μu.le
+        (c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := mul_le_mul_left' μu.le _
         _ = c * μ s + ε := by
           simp_rw [mul_add]
           rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
@@ -381,7 +381,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,
           Set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply]
       calc
-        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := ENNReal.mul_le_mul le_rfl μF.le
+        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := mul_le_mul_left' μF.le _
         _ = c * μ F + ε := by
           simp_rw [mul_add]
           rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]

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

@@ -75,7 +75,7 @@ See result `measure_theory.Lp.continuous_map_dense`, in the file
 -/
 
 
-open Ennreal NNReal
+open ENNReal NNReal
 
 open MeasureTheory MeasureTheory.Measure
 
@@ -104,7 +104,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
     by_cases h : (∫⁻ x, f x ∂μ) = ⊤
     · refine'
         ⟨fun x => c, fun x => _, lowerSemicontinuous_const, by
-          simp only [Ennreal.top_add, le_top, h]⟩
+          simp only [ENNReal.top_add, le_top, h]⟩
       simp only [simple_func.coe_const, simple_func.const_zero, simple_func.coe_zero,
         Set.piecewise_eq_indicator, simple_func.coe_piecewise]
       exact Set.indicator_le_self _ _ _
@@ -114,15 +114,15 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
         simp only [hc, Set.indicator_zero', Pi.zero_apply, simple_func.const_zero, imp_true_iff,
           eq_self_iff_true, simple_func.coe_zero, Set.piecewise_eq_indicator,
           simple_func.coe_piecewise, le_zero_iff]
-      · simp only [lintegral_const, zero_mul, zero_le, Ennreal.coe_zero]
+      · simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
     have : μ s < μ s + ε / c :=
       by
-      have : (0 : ℝ≥0∞) < ε / c := Ennreal.div_pos_iff.2 ⟨ε0, Ennreal.coe_ne_top⟩
-      simpa using Ennreal.add_lt_add_left _ this
+      have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
+      simpa using ENNReal.add_lt_add_left _ this
       simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, simple_func.coe_const,
-        Function.const_apply, lintegral_const, Ennreal.coe_indicator, Set.univ_inter,
-        Ennreal.coe_ne_top, MeasurableSet.univ, WithTop.mul_eq_top_iff, simple_func.const_zero,
-        or_false_iff, lintegral_indicator, Ennreal.coe_eq_zero, Ne.def, not_false_iff,
+        Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
+        ENNReal.coe_ne_top, MeasurableSet.univ, WithTop.mul_eq_top_iff, simple_func.const_zero,
+        or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
         simple_func.coe_zero, Set.piecewise_eq_indicator, simple_func.coe_piecewise, false_and_iff,
         restrict_apply] using h
     obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _)(_ : u ⊇ s), IsOpen u ∧ μ u < μ s + ε / c :=
@@ -134,26 +134,26 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
       exact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _
     · suffices (c : ℝ≥0∞) * μ u ≤ c * μ s + ε by
         simpa only [hs, u_open.measurable_set, simple_func.coe_const, Function.const_apply,
-          lintegral_const, Ennreal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
+          lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
           simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,
           Set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply]
       calc
-        (c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := Ennreal.mul_le_mul le_rfl μu.le
+        (c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := ENNReal.mul_le_mul le_rfl μu.le
         _ = c * μ s + ε := by
           simp_rw [mul_add]
-          rw [Ennreal.mul_div_cancel' _ Ennreal.coe_ne_top]
+          rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
           simpa using hc
         
-  · rcases h₁ (Ennreal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
-    rcases h₂ (Ennreal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
+  · rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
+    rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
     refine'
       ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, _⟩
-    simp only [simple_func.coe_add, Ennreal.coe_add, Pi.add_apply]
+    simp only [simple_func.coe_add, ENNReal.coe_add, Pi.add_apply]
     rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
       lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
     convert add_le_add g₁int g₂int using 1
     simp only
-    conv_lhs => rw [← Ennreal.add_halves ε]
+    conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
 
@@ -168,7 +168,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
     ∃ g : α → ℝ≥0∞,
       (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
   by
-  rcases Ennreal.exists_pos_sum_of_countable' εpos ℕ with ⟨δ, δpos, hδ⟩
+  rcases ENNReal.exists_pos_sum_of_countable' εpos ℕ with ⟨δ, δpos, hδ⟩
   have :
     ∀ n,
       ∃ g : α → ℝ≥0,
@@ -181,23 +181,23 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
   choose g f_le_g gcont hg using this
   refine' ⟨fun x => ∑' n, g n x, fun x => _, _, _⟩
   · rw [← tsum_eapprox_diff f hf]
-    exact Ennreal.tsum_le_tsum fun n => Ennreal.coe_le_coe.2 (f_le_g n x)
+    exact ENNReal.tsum_le_tsum fun n => ENNReal.coe_le_coe.2 (f_le_g n x)
   · apply lowerSemicontinuous_tsum fun n => _
     exact
       ennreal.continuous_coe.comp_lower_semicontinuous (gcont n) fun x y hxy =>
-        Ennreal.coe_le_coe.2 hxy
+        ENNReal.coe_le_coe.2 hxy
   ·
     calc
       (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
-        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_ennreal.AeMeasurable]
-      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := Ennreal.tsum_le_tsum hg
-      _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := Ennreal.tsum_add
+        rw [lintegral_tsum fun n => (gcont n).Measurable.coe_nNReal_eNNReal.AeMeasurable]
+      _ ≤ ∑' n, (∫⁻ x, eapprox_diff f n x ∂μ) + δ n := ENNReal.tsum_le_tsum hg
+      _ = (∑' n, ∫⁻ x, eapprox_diff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
         refine' add_le_add _ hδ.le
         rw [← lintegral_tsum]
         · simp_rw [tsum_eapprox_diff f hf, le_refl]
         · intro n
-          exact (simple_func.measurable _).coe_nNReal_ennreal.AeMeasurable
+          exact (simple_func.measurable _).coe_nNReal_eNNReal.AeMeasurable
       
 #align measure_theory.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
 
@@ -210,16 +210,16 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
     ∃ g : α → ℝ≥0∞,
       (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
   by
-  have : ε / 2 ≠ 0 := (Ennreal.half_pos ε0).ne'
+  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
   rcases exists_pos_lintegral_lt_of_sigma_finite μ this with ⟨w, wpos, wmeas, wint⟩
   let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
-  rcases exists_le_lower_semicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nNReal_ennreal
+  rcases exists_le_lower_semicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nNReal_eNNReal
       this with
     ⟨g, le_g, gcont, gint⟩
   refine' ⟨g, fun x => _, gcont, _⟩
   ·
     calc
-      (f x : ℝ≥0∞) < f' x := by simpa [← Ennreal.coe_lt_coe] using add_lt_add_left (wpos x) (f x)
+      (f x : ℝ≥0∞) < f' x := by simpa [← ENNReal.coe_lt_coe] using add_lt_add_left (wpos x) (f x)
       _ ≤ g x := le_g x
       
   ·
@@ -228,7 +228,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
       _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
         rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := add_le_add_right (add_le_add_left wint.le _) _
-      _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, Ennreal.add_halves]
+      _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
       
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
 
@@ -241,7 +241,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable [SigmaFinite
     ∃ g : α → ℝ≥0∞,
       (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε :=
   by
-  have : ε / 2 ≠ 0 := (Ennreal.half_pos ε0).ne'
+  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
   rcases exists_lt_lower_semicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
     ⟨g0, f_lt_g0, g0_cont, g0_int⟩
   rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
@@ -270,7 +270,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable [SigmaFinite
         · convert g1_int
           simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
             lintegral_indicator, mul_zero, restrict_apply]
-      _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, Ennreal.add_halves, zero_add]
+      _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
       
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aeMeasurable
 
@@ -299,33 +299,33 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nNReal [SigmaFinite μ] (f : 
   have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
     (has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral).Ne
   rcases exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable μ f fmeas
-      (Ennreal.coe_ne_zero.2 δpos.ne') with
+      (ENNReal.coe_ne_zero.2 δpos.ne') with
     ⟨g, f_lt_g, gcont, gint⟩
   have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
   have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
-  have Ig : (∫⁻ a : α, Ennreal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ :=
+  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ :=
     by
     apply lintegral_congr_ae
     filter_upwards [g_lt_top]with _ hx
-    simp only [hx.ne, Ennreal.ofReal_toReal, Ne.def, not_false_iff]
+    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
   refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
   · refine' ⟨gcont.measurable.ennreal_to_real.ae_measurable.ae_strongly_measurable, _⟩
-    simp only [has_finite_integral_iff_norm, Real.norm_eq_abs, abs_of_nonneg Ennreal.toReal_nonneg]
+    simp only [has_finite_integral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
     convert gint_ne.lt_top using 1
   · rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]
     ·
       calc
-        Ennreal.toReal (∫⁻ a : α, Ennreal.ofReal (g a).toReal ∂μ) =
-            Ennreal.toReal (∫⁻ a : α, g a ∂μ) :=
+        ENNReal.toReal (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) =
+            ENNReal.toReal (∫⁻ a : α, g a ∂μ) :=
           by congr 1
-        _ ≤ Ennreal.toReal ((∫⁻ a : α, f a ∂μ) + δ) :=
+        _ ≤ ENNReal.toReal ((∫⁻ a : α, f a ∂μ) + δ) :=
           by
-          apply Ennreal.toReal_mono _ gint
+          apply ENNReal.toReal_mono _ gint
           simpa using int_f_ne_top
-        _ = Ennreal.toReal (∫⁻ a : α, f a ∂μ) + δ := by
-          rw [Ennreal.toReal_add int_f_ne_top Ennreal.coe_ne_top, Ennreal.coe_toReal]
-        _ < Ennreal.toReal (∫⁻ a : α, f a ∂μ) + ε := add_lt_add_left hδε _
-        _ = (∫⁻ a : α, Ennreal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
+        _ = ENNReal.toReal (∫⁻ a : α, f a ∂μ) + δ := by
+          rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]
+        _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := add_lt_add_left hδε _
+        _ = (∫⁻ a : α, ENNReal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
         
     · apply Filter.eventually_of_forall fun x => _
       simp
@@ -358,15 +358,15 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
       ·
         simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,
           simple_func.const_zero, zero_add, zero_le', simple_func.coe_zero,
-          Set.piecewise_eq_indicator, Ennreal.coe_zero, simple_func.coe_piecewise, zero_le]
+          Set.piecewise_eq_indicator, ENNReal.coe_zero, simple_func.coe_piecewise, zero_le]
     have μs_lt_top : μ s < ∞ := by
       simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, simple_func.coe_const, or_false_iff,
-        lintegral_const, Ennreal.coe_indicator, Set.univ_inter, Ennreal.coe_ne_top,
+        lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,
         restrict_apply MeasurableSet.univ, WithTop.mul_eq_top_iff, simple_func.const_zero,
-        Function.const_apply, lintegral_indicator, Ennreal.coe_eq_zero, Ne.def, not_false_iff,
+        Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
         simple_func.coe_zero, Set.piecewise_eq_indicator, simple_func.coe_piecewise,
         false_and_iff] using int_f
-    have : (0 : ℝ≥0∞) < ε / c := Ennreal.div_pos_iff.2 ⟨ε0, Ennreal.coe_ne_top⟩
+    have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
     obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _)(_ : F ⊆ s), IsClosed F ∧ μ s < μ F + ε / c :=
       hs.exists_is_closed_lt_add μs_lt_top.ne this.ne'
     refine'
@@ -377,30 +377,30 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
       exact Set.indicator_le_indicator_of_subset Fs (fun x => zero_le _) _
     · suffices (c : ℝ≥0∞) * μ s ≤ c * μ F + ε by
         simpa only [hs, F_closed.measurable_set, simple_func.coe_const, Function.const_apply,
-          lintegral_const, Ennreal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
+          lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
           simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,
           Set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply]
       calc
-        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := Ennreal.mul_le_mul le_rfl μF.le
+        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := ENNReal.mul_le_mul le_rfl μF.le
         _ = c * μ F + ε := by
           simp_rw [mul_add]
-          rw [Ennreal.mul_div_cancel' _ Ennreal.coe_ne_top]
+          rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
           simpa using hc
         
   · have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by
       rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
-    rcases h₁ (Ennreal.add_ne_top.1 A).1 (Ennreal.half_pos ε0).ne' with
+    rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
       ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
-    rcases h₂ (Ennreal.add_ne_top.1 A).2 (Ennreal.half_pos ε0).ne' with
+    rcases h₂ (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos ε0).ne' with
       ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
     refine'
       ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, _⟩
-    simp only [simple_func.coe_add, Ennreal.coe_add, Pi.add_apply]
+    simp only [simple_func.coe_add, ENNReal.coe_add, Pi.add_apply]
     rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
       lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
     convert add_le_add g₁int g₂int using 1
     simp only
-    conv_lhs => rw [← Ennreal.add_halves ε]
+    conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le
 
@@ -415,28 +415,28 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
   obtain ⟨fs, fs_le_f, int_fs⟩ :
     ∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 :=
     by
-    have := Ennreal.lt_add_right int_f (Ennreal.half_pos ε0).ne'
+    have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [Ennreal.bsupr_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
+    erw [ENNReal.bsupr_add] at this <;> [skip, exact ⟨0, fun x => by simp⟩]
     simp only [lt_supᵢ_iff] at this
     rcases this with ⟨fs, fs_le_f, int_fs⟩
-    refine' ⟨fs, fun x => by simpa only [Ennreal.coe_le_coe] using fs_le_f x, _⟩
+    refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩
     convert int_fs.le
     rw [← simple_func.lintegral_eq_lintegral]
     rfl
   have int_fs_lt_top : (∫⁻ x, fs x ∂μ) ≠ ∞ :=
     by
     apply ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)
-    simpa only [Ennreal.coe_le_coe] using fs_le_f x
+    simpa only [ENNReal.coe_le_coe] using fs_le_f x
   obtain ⟨g, g_le_fs, gcont, gint⟩ :
     ∃ g : α → ℝ≥0,
       (∀ x, g x ≤ fs x) ∧ UpperSemicontinuous g ∧ (∫⁻ x, fs x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε / 2 :=
-    fs.exists_upper_semicontinuous_le_lintegral_le int_fs_lt_top (Ennreal.half_pos ε0).ne'
+    fs.exists_upper_semicontinuous_le_lintegral_le int_fs_lt_top (ENNReal.half_pos ε0).ne'
   refine' ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _⟩
   calc
     (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
     _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := add_le_add gint le_rfl
-    _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, Ennreal.add_halves]
+    _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
     
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
 
@@ -452,7 +452,7 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
           Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, g x ∂μ :=
   by
   lift ε to ℝ≥0 using εpos.le
-  rw [NNReal.coe_pos, ← Ennreal.coe_pos] at εpos
+  rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
   have If : (∫⁻ x, f x ∂μ) < ∞ := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral
   rcases exists_upper_semicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
   have Ig : (∫⁻ x, g x ∂μ) < ∞ :=
@@ -465,9 +465,9 @@ theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
     exact Filter.eventually_of_forall fun x => by simp [gf x]
   · rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]
     · rw [sub_le_iff_le_add]
-      convert Ennreal.toReal_mono _ gint
+      convert ENNReal.toReal_mono _ gint
       · simp
-      · rw [Ennreal.toReal_add Ig.ne Ennreal.coe_ne_top]
+      · rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]
         simp
       · simpa using Ig.ne
     · apply Filter.eventually_of_forall
@@ -520,13 +520,13 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
         integral_congr_ae ae_g
       _ = (∫ x : α, Ereal.toReal (gp x) ∂μ) - ∫ x : α, gm x ∂μ :=
         by
-        simp only [Ereal.toReal_coe_ennreal, Ennreal.coe_toReal, coe_coe]
+        simp only [Ereal.toReal_coe_eNNReal, ENNReal.coe_toReal, coe_coe]
         exact integral_sub gp_integrable gm_integrable
       _ < (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ∫ x : α, gm x ∂μ :=
         by
         apply sub_lt_sub_right
         convert gpint
-        simp only [Ereal.toReal_coe_ennreal]
+        simp only [Ereal.toReal_coe_eNNReal]
       _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, fm x ∂μ) - δ) := sub_le_sub_left gmint _
       _ = (∫ x : α, f x ∂μ) + 2 * δ :=
         by
@@ -539,34 +539,34 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   show ∀ᵐ x : α ∂μ, g x < ⊤
   · filter_upwards [gp_lt_top]with _ hx
     simp only [g, sub_eq_add_neg, coe_coe, Ne.def, (Ereal.add_lt_top _ _).Ne, lt_top_iff_ne_top,
-      lt_top_iff_ne_top.1 hx, Ereal.coe_ennreal_eq_top_iff, not_false_iff, Ereal.neg_eq_top_iff,
-      Ereal.coe_ennreal_ne_bot]
+      lt_top_iff_ne_top.1 hx, Ereal.coe_eNNReal_eq_top_iff, not_false_iff, Ereal.neg_eq_top_iff,
+      Ereal.coe_eNNReal_ne_bot]
   show ∀ x, (f x : Ereal) < g x
   · intro x
     rw [Ereal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]
     refine' Ereal.sub_lt_sub_of_lt_of_le _ _ _ _
-    · simp only [Ereal.coe_ennreal_lt_coe_ennreal_iff, coe_coe]
+    · simp only [Ereal.coe_eNNReal_lt_coe_eNNReal_iff, coe_coe]
       exact fp_lt_gp x
-    · simp only [Ennreal.coe_le_coe, Ereal.coe_ennreal_le_coe_ennreal_iff, coe_coe]
+    · simp only [ENNReal.coe_le_coe, Ereal.coe_eNNReal_le_coe_eNNReal_iff, coe_coe]
       exact gm_le_fm x
-    · simp only [Ereal.coe_ennreal_ne_bot, Ne.def, not_false_iff, coe_coe]
+    · simp only [Ereal.coe_eNNReal_ne_bot, Ne.def, not_false_iff, coe_coe]
     · simp only [Ereal.coe_nNReal_ne_top, Ne.def, not_false_iff, coe_coe]
   show LowerSemicontinuous g
   · apply LowerSemicontinuous.add'
     ·
       exact
         continuous_coe_ennreal_ereal.comp_lower_semicontinuous gpcont fun x y hxy =>
-          Ereal.coe_ennreal_le_coe_ennreal_iff.2 hxy
+          Ereal.coe_eNNReal_le_coe_eNNReal_iff.2 hxy
     · apply
         ereal.continuous_neg.comp_upper_semicontinuous_antitone _ fun x y hxy =>
           Ereal.neg_le_neg_iff.2 hxy
       dsimp
       apply
         continuous_coe_ennreal_ereal.comp_upper_semicontinuous _ fun x y hxy =>
-          Ereal.coe_ennreal_le_coe_ennreal_iff.2 hxy
+          Ereal.coe_eNNReal_le_coe_eNNReal_iff.2 hxy
       exact
         ennreal.continuous_coe.comp_upper_semicontinuous gmcont fun x y hxy =>
-          Ennreal.coe_le_coe.2 hxy
+          ENNReal.coe_le_coe.2 hxy
     · intro x
       exact Ereal.continuousAt_add (by simp) (by simp)
 #align measure_theory.exists_lt_lower_semicontinuous_integral_lt MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt

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

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

@@ -186,7 +186,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
   · calc
       ∫⁻ x, ∑' n : ℕ, g n x ∂μ = ∑' n, ∫⁻ x, g n x ∂μ := by
         rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]
-      _ ≤ ∑' n, ((∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n) := (ENNReal.tsum_le_tsum hg)
+      _ ≤ ∑' n, ((∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n) := ENNReal.tsum_le_tsum hg
       _ = ∑' n, ∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
         refine' add_le_add _ hδ.le
@@ -219,7 +219,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α →
       (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
       _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
         rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
-      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
+      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := add_le_add_right (add_le_add_left wint.le _) _
       _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
 
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
@@ -304,7 +304,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
           simpa using int_f_ne_top
         _ = ENNReal.toReal (∫⁻ a : α, f a ∂μ) + δ := by
           rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]
-        _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := (add_lt_add_left hδε _)
+        _ < ENNReal.toReal (∫⁻ a : α, f a ∂μ) + ε := add_lt_add_left hδε _
         _ = (∫⁻ a : α, ENNReal.ofReal ↑(f a) ∂μ).toReal + ε := by simp
 
     · apply Filter.eventually_of_forall fun x => _; simp
@@ -410,7 +410,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
   refine' ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _⟩
   calc
     (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
-    _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := (add_le_add gint le_rfl)
+    _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := add_le_add gint le_rfl
     _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
 
 #align measure_theory.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
@@ -494,7 +494,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
           apply sub_lt_sub_right
           convert gpint
           simp only [EReal.toReal_coe_ennreal]
-        _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, ↑(fm x) ∂μ) - δ) := (sub_le_sub_left gmint _)
+        _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, ↑(fm x) ∂μ) - δ) := sub_le_sub_left gmint _
         _ = (∫ x : α, f x ∂μ) + 2 * δ := by
           simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring
         _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]

@@ -118,7 +118,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
       simpa [f, hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
         Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
         ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
-        or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
+        or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
         SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,
         restrict_apply] using h
     obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _), u ⊇ s ∧ IsOpen u ∧ μ u < μ s + ε / c :=
@@ -289,7 +289,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
   have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
     apply lintegral_congr_ae
     filter_upwards [g_lt_top] with _ hx
-    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
+    simp only [hx.ne, ENNReal.ofReal_toReal, Ne, not_false_iff]
   refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
   · refine' ⟨gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
     simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
@@ -340,7 +340,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
       simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,
         lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,
         Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
-        Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
+        Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
         SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,
         false_and_iff] using int_f
     have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
@@ -501,7 +501,7 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
   case aelt =>
     show ∀ᵐ x : α ∂μ, g x < ⊤
     filter_upwards [gp_lt_top] with ?_ hx
-    simp only [g, sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,
+    simp only [g, sub_eq_add_neg, Ne, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,
       lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,
       EReal.coe_ennreal_ne_bot]
   case lt =>
@@ -512,8 +512,8 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
     · simp only [EReal.coe_ennreal_lt_coe_ennreal_iff]; exact fp_lt_gp x
     · simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff]
       exact gm_le_fm x
-    · simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff]
-    · simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff]
+    · simp only [EReal.coe_ennreal_ne_bot, Ne, not_false_iff]
+    · simp only [EReal.coe_nnreal_ne_top, Ne, not_false_iff]
   case lsc =>
     show LowerSemicontinuous g
     apply LowerSemicontinuous.add'

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.

@@ -278,7 +278,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
     convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
     simp only [Real.toNNReal_coe]
   lift ε to ℝ≥0 using εpos.le
-  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
+  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε := exists_between εpos
   have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
     (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
   rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -115,7 +115,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
       simpa using ENNReal.add_lt_add_left ?aux this
     case aux =>
       classical
-      simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
+      simpa [f, hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
         Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
         ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
         or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
@@ -497,11 +497,11 @@ theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → 
         _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, ↑(fm x) ∂μ) - δ) := (sub_le_sub_left gmint _)
         _ = (∫ x : α, f x ∂μ) + 2 * δ := by
           simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring
-        _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [mul_comm]
+        _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [δ, mul_comm]
   case aelt =>
     show ∀ᵐ x : α ∂μ, g x < ⊤
     filter_upwards [gp_lt_top] with ?_ hx
-    simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,
+    simp only [g, sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,
       lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,
       EReal.coe_ennreal_ne_bot]
   case lt =>

@@ -62,8 +62,8 @@ Finally, we glue them together to obtain the main statement
 ## Related results
 
 Are you looking for a result on approximation by continuous functions (not just semicontinuous)?
-See result `measure_theory.Lp.continuous_map_dense`, in the file
-`measure_theory.continuous_map_dense`.
+See result `MeasureTheory.Lp.boundedContinuousFunction_dense`, in the file
+`Mathlib/MeasureTheory/Function/ContinuousMapDense.lean`.
 
 ## References
 
@@ -81,7 +81,6 @@ variable {α : Type*} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α]
 
 namespace MeasureTheory
 
--- mathport name: «expr →ₛ »
 local infixr:25 " →ₛ " => SimpleFunc
 
 /-! ### Lower semicontinuous upper bound for nonnegative functions -/
@@ -400,7 +399,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩
     convert int_fs.le
     rw [← SimpleFunc.lintegral_eq_lintegral]
-    rfl
+    simp only [SimpleFunc.coe_map, Function.comp_apply]
   have int_fs_lt_top : (∫⁻ x, fs x ∂μ) ≠ ∞ := by
     refine' ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)
     simpa only [ENNReal.coe_le_coe] using fs_le_f x

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -110,7 +110,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
         simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
           eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
           SimpleFunc.coe_piecewise, le_zero_iff]
-      · simp only [lintegral_const, MulZeroClass.zero_mul, zero_le, ENNReal.coe_zero]
+      · simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
     have : μ s < μ s + ε / c := by
       have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
       simpa using ENNReal.add_lt_add_left ?aux this
@@ -257,7 +257,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
           exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
         · convert g1_int
           simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
-            lintegral_indicator, MulZeroClass.mul_zero, restrict_apply]
+            lintegral_indicator, mul_zero, restrict_apply]
       _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]
 
 #align measure_theory.exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
@@ -333,7 +333,7 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
           SimpleFunc.coe_piecewise, le_zero_iff]
       · classical
-        simp only [hc, Set.indicator_zero', lintegral_const, MulZeroClass.zero_mul, Pi.zero_apply,
+        simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,
           SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,
           Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]
     have μs_lt_top : μ s < ∞ := by

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -149,7 +149,6 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
     rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
       lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
     convert add_le_add g₁int g₂int using 1
-    simp only
     conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
@@ -378,7 +377,6 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
     rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
       lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
     convert add_le_add g₁int g₂int using 1
-    simp only
     conv_lhs => rw [← ENNReal.add_halves ε]
     abel
 #align measure_theory.simple_func.exists_upper_semicontinuous_le_lintegral_le MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le

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

This has nice performance benefits.

@@ -76,7 +76,7 @@ open scoped ENNReal NNReal
 
 open MeasureTheory MeasureTheory.Measure
 
-variable {α : Type _} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : Measure α)
+variable {α : Type*} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : Measure α)
   [WeaklyRegular μ]
 
 namespace MeasureTheory

• 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 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module measure_theory.integral.vitali_caratheodory
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
-! 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.Topology.Semicontinuous
 import Mathlib.MeasureTheory.Integral.Bochner
 import Mathlib.Topology.Instances.EReal
 
+#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
+
 /-!
 # Vitali-Carathéodory theorem
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -399,7 +399,7 @@ theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f :
     -- Porting note: need to name identifier (not `this`), because `conv_rhs at this` errors
     have aux := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
     conv_rhs at aux => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
-    erw [ENNReal.biSup_add] at aux  <;> [skip; exact ⟨0, fun x => by simp⟩]
+    erw [ENNReal.biSup_add] at aux <;> [skip; exact ⟨0, fun x => by simp⟩]
     simp only [lt_iSup_iff] at aux
     rcases aux with ⟨fs, fs_le_f, int_fs⟩
     refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

@@ -249,7 +249,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
     · have := le_g1 x
       simp only [h, Set.indicator_of_mem, top_le_iff] at this
       simp [this]
-    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs ; exact hs h
+    · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
       rw [this]
       exact (f_lt_g0 x).trans_le le_self_add
   · calc

This PR is the result of running

find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;

which firstly replaces . focusing dots with · and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.

@@ -252,8 +252,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
     · have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs ; exact hs h
       rw [this]
       exact (f_lt_g0 x).trans_le le_self_add
-  ·
-    calc
+  · calc
       ∫⁻ x, g0 x + g1 x ∂μ = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
         lintegral_add_left g0_cont.measurable _
       _ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) := by
@@ -301,8 +300,7 @@ theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : 
     simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
     convert gint_ne.lt_top using 1
   · rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]
-    ·
-      calc
+    · calc
         ENNReal.toReal (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) =
             ENNReal.toReal (∫⁻ a : α, g a ∂μ) :=
           by congr 1

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

@@ -191,8 +191,8 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
   · calc
       ∫⁻ x, ∑' n : ℕ, g n x ∂μ = ∑' n, ∫⁻ x, g n x ∂μ := by
         rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]
-      _ ≤ ∑' n, (∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
-      _ = (∑' n, ∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
+      _ ≤ ∑' n, ((∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n) := (ENNReal.tsum_le_tsum hg)
+      _ = ∑' n, ∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ + ∑' n, δ n := ENNReal.tsum_add
       _ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
         refine' add_le_add _ hδ.le
         rw [← lintegral_tsum]

@@ -100,7 +100,7 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
       (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε := by
   induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
   · let f := SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)
-    by_cases h : (∫⁻ x, f x ∂μ) = ⊤
+    by_cases h : ∫⁻ x, f x ∂μ = ⊤
     · refine'
         ⟨fun _ => c, fun x => _, lowerSemicontinuous_const, by
           simp only [_root_.top_add, le_top, h]⟩
@@ -189,7 +189,7 @@ theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf :
       ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy =>
         ENNReal.coe_le_coe.2 hxy
   · calc
-      (∫⁻ x, ∑' n : ℕ, g n x ∂μ) = ∑' n, ∫⁻ x, g n x ∂μ := by
+      ∫⁻ x, ∑' n : ℕ, g n x ∂μ = ∑' n, ∫⁻ x, g n x ∂μ := by
         rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]
       _ ≤ ∑' n, (∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n := (ENNReal.tsum_le_tsum hg)
       _ = (∑' n, ∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + ∑' n, δ n := ENNReal.tsum_add
@@ -254,7 +254,7 @@ theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite
       exact (f_lt_g0 x).trans_le le_self_add
   ·
     calc
-      (∫⁻ x, g0 x + g1 x ∂μ) = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
+      ∫⁻ x, g0 x + g1 x ∂μ = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
         lintegral_add_left g0_cont.measurable _
       _ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) := by
         refine' add_le_add _ _