measure_theory.integral.layercake ⟷ Mathlib.MeasureTheory.Integral.Layercake

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

@@ -275,7 +275,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   simp_rw [g_def] at key
   rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
-    rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
+    rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel₀ (f ω ^ p) p_pos.ne.symm]
   · exact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
 -/

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

@@ -120,7 +120,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite (μ
         simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
           mul_one]
       · have h_copy := h
-        simp only [mem_Ioc, not_and, not_le] at h 
+        simp only [mem_Ioc, not_and, not_le] at h
         by_cases h' : 0 < s
         ·
           simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
@@ -162,7 +162,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite (μ
       rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]
   rw [aux₂]
   have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
-  simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble 
+  simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble
   exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AEMeasurable.indicator mble
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
 -/
@@ -236,7 +236,7 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
   have key :=
     lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
       (eventually_of_forall fun t => zero_le_one)
-  simp_rw [def_cst, ENNReal.ofReal_one, mul_one] at key 
+  simp_rw [def_cst, ENNReal.ofReal_one, mul_one] at key
   rw [← key]
   congr with ω
   simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]
@@ -272,7 +272,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
     intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
-  simp_rw [g_def] at key 
+  simp_rw [g_def] at key
   rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
     rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
@@ -303,7 +303,7 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
     ext a
     simp only [mem_set_of_eq, mem_union]
     apply le_iff_lt_or_eq
-  rw [show {a : α | t = g a} = {a : α | g a = t} by simp_rw [eq_comm]] at uni 
+  rw [show {a : α | t = g a} = {a : α | g a = t} by simp_rw [eq_comm]] at uni
   have disj : {a : α | t < g a} ∩ {a : α | g a = t} = ∅ :=
     by
     ext a
@@ -314,8 +314,8 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
       measure_union (disjoint_iff_inter_eq_empty.mpr disj)
         (g_mble (finite.measurable_set (finite_singleton t)))]
   by_contra con
-  rw [not_lt, nonpos_iff_eq_zero] at con 
-  rw [Con, add_zero] at μ_add 
+  rw [not_lt, nonpos_iff_eq_zero] at con
+  rw [Con, add_zero] at μ_add
   exact ht μ_add
 #align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
 

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

@@ -268,7 +268,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : ℝ)))]
     intro t t_pos
     rw [g_def]
-    exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)
+    exact Real.rpow_nonneg (mem_Ioi.mp t_pos).le (p - 1)
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
     intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn

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

@@ -73,7 +73,7 @@ namespace MeasureTheory
 
 variable {α : Type _} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
-#print MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable /-
+#print MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite /-
 /-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
 formula), with a measurability assumption that would also essentially follow from the
 integrability assumptions.
@@ -81,9 +81,10 @@ integrability assumptions.
 See `measure_theory.lintegral_comp_eq_lintegral_meas_le_mul` and
 `measure_theory.lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
 cake formula. -/
-theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α) [SigmaFinite μ]
-    (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
-    (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t) :
+theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite (μ : Measure α)
+    [SigmaFinite μ] (f_nn : 0 ≤ f) (f_mble : Measurable f)
+    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
+    (g_nn : ∀ t > 0, 0 ≤ g t) :
     ∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
   by
@@ -163,7 +164,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
   have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
   simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble 
   exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AEMeasurable.indicator mble
-#align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
+#align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
 -/
 
 #print MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul /-
@@ -293,8 +294,7 @@ variable {β : Type _} [MeasurableSpace β] [MeasurableSingletonClass β]
 
 namespace Measure
 
-#print Measure.meas_eq_pos_of_meas_le_ne_meas_lt /-
-theorem meas_eq_pos_of_meas_le_ne_meas_lt {R : Type _} [LinearOrder R] [MeasurableSpace R]
+theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
     (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} :=
   by
@@ -317,31 +317,26 @@ theorem meas_eq_pos_of_meas_le_ne_meas_lt {R : Type _} [LinearOrder R] [Measurab
   rw [not_lt, nonpos_iff_eq_zero] at con 
   rw [Con, add_zero] at μ_add 
   exact ht μ_add
-#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_eq_pos_of_meas_le_ne_meas_lt
--/
+#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
 
-#print Measure.countable_meas_le_ne_meas_lt /-
 theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
     {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
-  Countable.mono (show _ from fun t ht => meas_eq_pos_of_meas_le_ne_meas_lt μ g_mble ht)
+  Countable.mono (show _ from fun t ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
     (Measure.countable_meas_level_set_pos g_mble)
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
--/
 
-#print Measure.meas_le_ae_eq_meas_lt /-
 theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
     (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
   Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _
 #align measure.meas_le_ae_eq_meas_lt Measure.meas_le_ae_eq_meas_lt
--/
 
 end Measure
 
 variable {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
-#print lintegral_comp_eq_lintegral_meas_lt_mul /-
+#print MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul /-
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
 
 Let `f` be a non-negative measurable function on a sigma-finite measure space. Let `G` be an
@@ -354,8 +349,8 @@ Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0 .. 
 
 See `lintegral_comp_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
+theorem MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ]
+    (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     ∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) :=
@@ -364,10 +359,10 @@ theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
   apply lintegral_congr_ae
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
-#align lintegral_comp_eq_lintegral_meas_lt_mul lintegral_comp_eq_lintegral_meas_lt_mul
+#align lintegral_comp_eq_lintegral_meas_lt_mul MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul
 -/
 
-#print lintegral_eq_lintegral_meas_lt /-
+#print MeasureTheory.lintegral_eq_lintegral_meas_lt /-
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -375,17 +370,17 @@ be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0 .. ∞, μ {ω
 
 See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
+theorem MeasureTheory.lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) : ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t < f a} :=
   by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
-#align lintegral_eq_lintegral_meas_lt lintegral_eq_lintegral_meas_lt
+#align lintegral_eq_lintegral_meas_lt MeasureTheory.lintegral_eq_lintegral_meas_lt
 -/
 
-#print lintegral_rpow_eq_lintegral_meas_lt_mul /-
+#print MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul /-
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -393,8 +388,8 @@ be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0 .. ∞,
 
 See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
+theorem MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ]
+    (f_nn : 0 ≤ f) (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
     ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) :=
   by
@@ -403,7 +398,7 @@ theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
   apply lintegral_congr_ae
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
-#align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
+#align lintegral_rpow_eq_lintegral_meas_lt_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul
 -/
 
 end LayercakeLt

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Kalle Kytölä. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
 -/
-import Mathbin.MeasureTheory.Integral.IntervalIntegral
-import Mathbin.Analysis.SpecialFunctions.Integrals
+import MeasureTheory.Integral.IntervalIntegral
+import Analysis.SpecialFunctions.Integrals
 
 #align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -293,8 +293,8 @@ variable {β : Type _} [MeasurableSpace β] [MeasurableSingletonClass β]
 
 namespace Measure
 
-#print Measure.meas_le_ne_meas_lt_subset_meas_pos /-
-theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [MeasurableSpace R]
+#print Measure.meas_eq_pos_of_meas_le_ne_meas_lt /-
+theorem meas_eq_pos_of_meas_le_ne_meas_lt {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
     (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} :=
   by
@@ -317,14 +317,14 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
   rw [not_lt, nonpos_iff_eq_zero] at con 
   rw [Con, add_zero] at μ_add 
   exact ht μ_add
-#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
+#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_eq_pos_of_meas_le_ne_meas_lt
 -/
 
 #print Measure.countable_meas_le_ne_meas_lt /-
 theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
     {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
-  Countable.mono (show _ from fun t ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
+  Countable.mono (show _ from fun t ht => meas_eq_pos_of_meas_le_ne_meas_lt μ g_mble ht)
     (Measure.countable_meas_level_set_pos g_mble)
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
 -/

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Kalle Kytölä. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
-
-! This file was ported from Lean 3 source module measure_theory.integral.layercake
-! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Integral.IntervalIntegral
 import Mathbin.Analysis.SpecialFunctions.Integrals
 
+#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
 /-!
 # The layer cake formula / Cavalieri's principle / tail probability formula
 

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

@@ -76,6 +76,7 @@ namespace MeasureTheory
 
 variable {α : Type _} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
+#print MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable /-
 /-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
 formula), with a measurability assumption that would also essentially follow from the
 integrability assumptions.
@@ -166,7 +167,9 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
   simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble 
   exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AEMeasurable.indicator mble
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
+-/
 
+#print MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul /-
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
 
 Let `f` be a non-negative measurable function on a sigma-finite measure space. Let `G` be an
@@ -216,7 +219,9 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     lintegral_comp_eq_lintegral_meas_le_mul_of_measurable μ f_nn f_mble G_intble G_mble
       fun t t_pos => G_nn t
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul
+-/
 
+#print MeasureTheory.lintegral_eq_lintegral_meas_le /-
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -238,7 +243,9 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
   congr with ω
   simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]
 #align measure_theory.lintegral_eq_lintegral_meas_le MeasureTheory.lintegral_eq_lintegral_meas_le
+-/
 
+#print MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul /-
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -273,6 +280,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
   · exact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
+-/
 
 end MeasureTheory
 
@@ -288,6 +296,7 @@ variable {β : Type _} [MeasurableSpace β] [MeasurableSingletonClass β]
 
 namespace Measure
 
+#print Measure.meas_le_ne_meas_lt_subset_meas_pos /-
 theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
     (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} :=
@@ -312,24 +321,30 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
   rw [Con, add_zero] at μ_add 
   exact ht μ_add
 #align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
+-/
 
+#print Measure.countable_meas_le_ne_meas_lt /-
 theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
     {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
   Countable.mono (show _ from fun t ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
     (Measure.countable_meas_level_set_pos g_mble)
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
+-/
 
+#print Measure.meas_le_ae_eq_meas_lt /-
 theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
     (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
   Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _
 #align measure.meas_le_ae_eq_meas_lt Measure.meas_le_ae_eq_meas_lt
+-/
 
 end Measure
 
 variable {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
+#print lintegral_comp_eq_lintegral_meas_lt_mul /-
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
 
 Let `f` be a non-negative measurable function on a sigma-finite measure space. Let `G` be an
@@ -353,7 +368,9 @@ theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_comp_eq_lintegral_meas_lt_mul lintegral_comp_eq_lintegral_meas_lt_mul
+-/
 
+#print lintegral_eq_lintegral_meas_lt /-
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -369,7 +386,9 @@ theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_eq_lintegral_meas_lt lintegral_eq_lintegral_meas_lt
+-/
 
+#print lintegral_rpow_eq_lintegral_meas_lt_mul /-
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
@@ -388,6 +407,7 @@ theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
+-/
 
 end LayercakeLt
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
 
 ! This file was ported from Lean 3 source module measure_theory.integral.layercake
-! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.SpecialFunctions.Integrals
 /-!
 # The layer cake formula / Cavalieri's principle / tail probability formula
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the following layer cake formula.
 
 Consider a non-negative measurable function `f` on a sigma-finite measure space. Apply pointwise

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

@@ -83,7 +83,7 @@ cake formula. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α) [SigmaFinite μ]
     (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
   by
   have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t :=
@@ -179,7 +179,7 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
   by
   have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G :=
@@ -195,13 +195,13 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrable_on_empty
   have eq₁ :
-    (∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)) =
+    ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (G t) :=
     by
     apply lintegral_congr_ae
     filter_upwards [g_eq_G] with a ha
     rw [ha]
-  have eq₂ : ∀ ω, (∫ t in 0 ..f ω, g t) = ∫ t in 0 ..f ω, G t :=
+  have eq₂ : ∀ ω, ∫ t in 0 ..f ω, g t = ∫ t in 0 ..f ω, G t :=
     by
     refine' fun ω => intervalIntegral.integral_congr_ae _
     have fω_nn : 0 ≤ f ω := f_nn ω
@@ -222,8 +222,7 @@ be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0 .. ∞, μ {ω
 See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} :=
+    (f_mble : Measurable f) : ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} :=
   by
   set cst := fun t : ℝ => (1 : ℝ) with def_cst
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
@@ -246,11 +245,11 @@ See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   have one_lt_p : -1 < p - 1 := by linarith
-  have obs : ∀ x : ℝ, (∫ t : ℝ in 0 ..x, t ^ (p - 1)) = x ^ p / p :=
+  have obs : ∀ x : ℝ, ∫ t : ℝ in 0 ..x, t ^ (p - 1) = x ^ p / p :=
     by
     intro x
     rw [integral_rpow (Or.inl one_lt_p)]
@@ -343,7 +342,7 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) :=
   by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
@@ -360,8 +359,7 @@ be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0 .. ∞, μ {ω
 See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
 theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t < f a} :=
+    (f_mble : Measurable f) : ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t < f a} :=
   by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
@@ -378,7 +376,7 @@ See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]

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

@@ -84,7 +84,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t) :=
+      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
   by
   have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t :=
     by
@@ -97,8 +97,8 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     by
     intro ω
     have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
-      filter_upwards [self_mem_ae_restrict
-          (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))]with x hx using g_nn x hx.1
+      filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))] with x
+        hx using g_nn x hx.1
     rw [← of_real_integral_eq_lintegral_of_real (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]
     congr
     exact intervalIntegral.integral_of_le (f_nn ω)
@@ -133,30 +133,30 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
       by_cases s ∈ Ioi (0 : ℝ) <;> · simp [h]
     simp_rw [show
         (fun a => (Ici s).indicator (fun t : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
-          { a : α | s ≤ f a }.indicator (fun _ => 1) a
+          {a : α | s ≤ f a}.indicator (fun _ => 1) a
         by funext a; by_cases s ≤ f a <;> simp [h]]
     rw [lintegral_indicator]
     swap; · exact f_mble measurableSet_Ici
     rw [lintegral_one, measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,
       mul_assoc,
       show
-        (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ { a : α | s ≤ f a } =
-          (Ioi 0).indicator (fun _x : ℝ => 1 * μ { a : α | s ≤ f a }) s
+        (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ {a : α | s ≤ f a} =
+          (Ioi 0).indicator (fun _x : ℝ => 1 * μ {a : α | s ≤ f a}) s
         by by_cases 0 < s <;> simp [h]]
     simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]
     rfl
   have aux₂ :
     (Function.uncurry fun (x : α) (y : ℝ) =>
         (Ioc 0 (f x)).indicator (fun t : ℝ => ENNReal.ofReal (g t)) y) =
-      { p : α × ℝ | p.2 ∈ Ioc 0 (f p.1) }.indicator fun p => ENNReal.ofReal (g p.2) :=
+      {p : α × ℝ | p.2 ∈ Ioc 0 (f p.1)}.indicator fun p => ENNReal.ofReal (g p.2) :=
     by
     funext p
     cases p
     rw [Function.uncurry_apply_pair]
     by_cases p_snd ∈ Ioc 0 (f p_fst)
-    · have h' : (p_fst, p_snd) ∈ { p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst) } := h
+    · have h' : (p_fst, p_snd) ∈ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h
       rw [Set.indicator_of_mem h', Set.indicator_of_mem h]
-    · have h' : (p_fst, p_snd) ∉ { p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst) } := h
+    · have h' : (p_fst, p_snd) ∉ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h
       rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]
   rw [aux₂]
   have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
@@ -180,7 +180,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t) :=
+      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
   by
   have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G :=
     by
@@ -195,11 +195,11 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrable_on_empty
   have eq₁ :
-    (∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t)) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (G t) :=
+    (∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)) =
+      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (G t) :=
     by
     apply lintegral_congr_ae
-    filter_upwards [g_eq_G]with a ha
+    filter_upwards [g_eq_G] with a ha
     rw [ha]
   have eq₂ : ∀ ω, (∫ t in 0 ..f ω, g t) = ∫ t in 0 ..f ω, G t :=
     by
@@ -223,7 +223,7 @@ See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } :=
+    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} :=
   by
   set cst := fun t : ℝ => (1 : ℝ) with def_cst
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
@@ -247,7 +247,7 @@ instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
-      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (t ^ (p - 1)) :=
+      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   have one_lt_p : -1 < p - 1 := by linarith
   have obs : ∀ x : ℝ, (∫ t : ℝ in 0 ..x, t ^ (p - 1)) = x ^ p / p :=
@@ -288,20 +288,20 @@ namespace Measure
 
 theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
-    (ht : μ { a : α | t ≤ g a } ≠ μ { a : α | t < g a }) : 0 < μ { a : α | g a = t } :=
+    (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} :=
   by
-  have uni : { a : α | t ≤ g a } = { a : α | t < g a } ∪ { a : α | t = g a } :=
+  have uni : {a : α | t ≤ g a} = {a : α | t < g a} ∪ {a : α | t = g a} :=
     by
     ext a
     simp only [mem_set_of_eq, mem_union]
     apply le_iff_lt_or_eq
-  rw [show { a : α | t = g a } = { a : α | g a = t } by simp_rw [eq_comm]] at uni 
-  have disj : { a : α | t < g a } ∩ { a : α | g a = t } = ∅ :=
+  rw [show {a : α | t = g a} = {a : α | g a = t} by simp_rw [eq_comm]] at uni 
+  have disj : {a : α | t < g a} ∩ {a : α | g a = t} = ∅ :=
     by
     ext a
     simp only [mem_inter_iff, mem_set_of_eq, mem_empty_iff_false, iff_false_iff, not_and]
     exact ne_of_gt
-  have μ_add : μ { a : α | t ≤ g a } = μ { a : α | t < g a } + μ { a : α | g a = t } := by
+  have μ_add : μ {a : α | t ≤ g a} = μ {a : α | t < g a} + μ {a : α | g a = t} := by
     rw [uni,
       measure_union (disjoint_iff_inter_eq_empty.mpr disj)
         (g_mble (finite.measurable_set (finite_singleton t)))]
@@ -313,14 +313,14 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
 
 theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
-    { t : R | μ { a : α | t ≤ g a } ≠ μ { a : α | t < g a } }.Countable :=
+    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
   Countable.mono (show _ from fun t ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
     (Measure.countable_meas_level_set_pos g_mble)
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
 
 theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
-    (fun t => μ { a : α | t ≤ g a }) =ᵐ[ν] fun t => μ { a : α | t < g a } :=
+    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
   Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _
 #align measure.meas_le_ae_eq_meas_lt Measure.meas_le_ae_eq_meas_lt
 
@@ -344,11 +344,11 @@ theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t < f a } * ENNReal.ofReal (g t) :=
+      ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) :=
   by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble]with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_comp_eq_lintegral_meas_lt_mul lintegral_comp_eq_lintegral_meas_lt_mul
 
@@ -361,11 +361,11 @@ See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t < f a } :=
+    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t < f a} :=
   by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble]with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_eq_lintegral_meas_lt lintegral_eq_lintegral_meas_lt
 
@@ -379,12 +379,12 @@ instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
-      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t < f a } * ENNReal.ofReal (t ^ (p - 1)) :=
+      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
   apply congr_arg fun z => ENNReal.ofReal p * z
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble]with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
   rw [ht]
 #align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
 

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

@@ -118,7 +118,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
         simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
           mul_one]
       · have h_copy := h
-        simp only [mem_Ioc, not_and, not_le] at h
+        simp only [mem_Ioc, not_and, not_le] at h 
         by_cases h' : 0 < s
         ·
           simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
@@ -160,7 +160,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
       rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]
   rw [aux₂]
   have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
-  simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble
+  simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble 
   exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AEMeasurable.indicator mble
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
 
@@ -231,7 +231,7 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
   have key :=
     lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
       (eventually_of_forall fun t => zero_le_one)
-  simp_rw [def_cst, ENNReal.ofReal_one, mul_one] at key
+  simp_rw [def_cst, ENNReal.ofReal_one, mul_one] at key 
   rw [← key]
   congr with ω
   simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]
@@ -265,7 +265,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
     intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
-  simp_rw [g_def] at key
+  simp_rw [g_def] at key 
   rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
     rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
@@ -295,7 +295,7 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
     ext a
     simp only [mem_set_of_eq, mem_union]
     apply le_iff_lt_or_eq
-  rw [show { a : α | t = g a } = { a : α | g a = t } by simp_rw [eq_comm]] at uni
+  rw [show { a : α | t = g a } = { a : α | g a = t } by simp_rw [eq_comm]] at uni 
   have disj : { a : α | t < g a } ∩ { a : α | g a = t } = ∅ :=
     by
     ext a
@@ -306,8 +306,8 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
       measure_union (disjoint_iff_inter_eq_empty.mpr disj)
         (g_mble (finite.measurable_set (finite_singleton t)))]
   by_contra con
-  rw [not_lt, nonpos_iff_eq_zero] at con
-  rw [Con, add_zero] at μ_add
+  rw [not_lt, nonpos_iff_eq_zero] at con 
+  rw [Con, add_zero] at μ_add 
   exact ht μ_add
 #align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
 

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

@@ -60,7 +60,7 @@ layer cake representation, Cavalieri's principle, tail probability formula
 
 noncomputable section
 
-open ENNReal MeasureTheory
+open scoped ENNReal MeasureTheory
 
 open Set MeasureTheory Filter
 

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

@@ -128,18 +128,15 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
             MulZeroClass.zero_mul, mem_Ioc, false_and_iff]
     simp_rw [aux₁]
     rw [lintegral_const_mul']
-    swap
+    swap;
     · apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top
       by_cases s ∈ Ioi (0 : ℝ) <;> · simp [h]
     simp_rw [show
         (fun a => (Ici s).indicator (fun t : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
           { a : α | s ≤ f a }.indicator (fun _ => 1) a
-        by
-        funext a
-        by_cases s ≤ f a <;> simp [h]]
+        by funext a; by_cases s ≤ f a <;> simp [h]]
     rw [lintegral_indicator]
-    swap
-    · exact f_mble measurableSet_Ici
+    swap; · exact f_mble measurableSet_Ici
     rw [lintegral_one, measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,
       mul_assoc,
       show

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

@@ -272,7 +272,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
     rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
-  · exact ((f_mble.pow measurable_const).div_const p).eNNReal_ofReal
+  · exact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
 
 end MeasureTheory

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

@@ -188,7 +188,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G :=
     by
     refine' AEMeasurable.exists_measurable_nonneg _ g_nn
-    exact aEMeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.AEMeasurable
+    exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.AEMeasurable
   rcases ex_G with ⟨G, G_mble, G_nn, g_eq_G⟩
   have g_eq_G_on : ∀ t, g =ᵐ[volume.restrict (Ioc 0 t)] G := fun t =>
     ae_mono (measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G

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

@@ -322,8 +322,7 @@ theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
 
 theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R] [MeasurableSpace R]
-    [MeasurableSingletonClass R] (ν : Measure R) [HasNoAtoms ν] {g : α → R}
-    (g_mble : Measurable g) :
+    [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
     (fun t => μ { a : α | t ≤ g a }) =ᵐ[ν] fun t => μ { a : α | t < g a } :=
   Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _
 #align measure.meas_le_ae_eq_meas_lt Measure.meas_le_ae_eq_meas_lt

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

@@ -164,7 +164,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
   rw [aux₂]
   have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
   simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble
-  exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AeMeasurable.indicator mble
+  exact (ennreal.measurable_of_real.comp (g_mble.comp measurable_snd)).AEMeasurable.indicator mble
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
 
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
@@ -187,14 +187,14 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   by
   have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G :=
     by
-    refine' AeMeasurable.exists_measurable_nonneg _ g_nn
-    exact aeMeasurableIoiOfForallIoc fun t ht => (g_intble t ht).1.1.AeMeasurable
+    refine' AEMeasurable.exists_measurable_nonneg _ g_nn
+    exact aEMeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.AEMeasurable
   rcases ex_G with ⟨G, G_mble, G_nn, g_eq_G⟩
   have g_eq_G_on : ∀ t, g =ᵐ[volume.restrict (Ioc 0 t)] G := fun t =>
     ae_mono (measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G
   have G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t :=
     by
-    refine' fun t t_pos => ⟨(g_intble t t_pos).1.congrFunAe (g_eq_G_on t), _⟩
+    refine' fun t t_pos => ⟨(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _⟩
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrable_on_empty
   have eq₁ :
@@ -229,7 +229,8 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } :=
   by
   set cst := fun t : ℝ => (1 : ℝ) with def_cst
-  have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ => intervalIntegrableConst
+  have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
+    intervalIntegrable_const
   have key :=
     lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
       (eventually_of_forall fun t => zero_le_one)
@@ -265,7 +266,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     rw [g_def]
     exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
-    intervalIntegral.intervalIntegrableRpow' one_lt_p
+    intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
   simp_rw [g_def] at key
   rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
 
 ! This file was ported from Lean 3 source module measure_theory.integral.layercake
-! leanprover-community/mathlib commit 9003f28797c0664a49e4179487267c494477d853
+! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -194,7 +194,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     ae_mono (measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G
   have G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t :=
     by
-    refine' fun t t_pos => ⟨integrable_on.congr_fun' (g_intble t t_pos).1 (g_eq_G_on t), _⟩
+    refine' fun t t_pos => ⟨(g_intble t t_pos).1.congrFunAe (g_eq_G_on t), _⟩
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrable_on_empty
   have eq₁ :

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

@@ -120,10 +120,12 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
       · have h_copy := h
         simp only [mem_Ioc, not_and, not_le] at h
         by_cases h' : 0 < s
-        · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]
+        ·
+          simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
+            MulZeroClass.mul_zero]
         · have : s ∉ Ioi (0 : ℝ) := h'
-          simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero, zero_mul, mem_Ioc,
-            false_and_iff]
+          simp only [this, h', indicator_of_not_mem, not_false_iff, MulZeroClass.mul_zero,
+            MulZeroClass.zero_mul, mem_Ioc, false_and_iff]
     simp_rw [aux₁]
     rw [lintegral_const_mul']
     swap

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

@@ -60,7 +60,7 @@ layer cake representation, Cavalieri's principle, tail probability formula
 
 noncomputable section
 
-open Ennreal MeasureTheory
+open ENNReal MeasureTheory
 
 open Set MeasureTheory Filter
 
@@ -83,8 +83,8 @@ cake formula. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α) [SigmaFinite μ]
     (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t) :
-    (∫⁻ ω, Ennreal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * Ennreal.ofReal (g t) :=
+    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t) :=
   by
   have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t :=
     by
@@ -93,7 +93,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     · simp [← h]
     · exact g_intble t h
   have integrand_eq :
-    ∀ ω, Ennreal.ofReal (∫ t in 0 ..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), Ennreal.ofReal (g t) :=
+    ∀ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) :=
     by
     intro ω
     have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
@@ -102,14 +102,14 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     rw [← of_real_integral_eq_lintegral_of_real (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]
     congr
     exact intervalIntegral.integral_of_le (f_nn ω)
-  simp_rw [integrand_eq, ← lintegral_indicator (fun t => Ennreal.ofReal (g t)) measurableSet_Ioc, ←
+  simp_rw [integrand_eq, ← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc, ←
     lintegral_indicator _ measurableSet_Ioi]
   rw [lintegral_lintegral_swap]
   · apply congr_arg
     funext s
     have aux₁ :
-      (fun x => (Ioc 0 (f x)).indicator (fun t : ℝ => Ennreal.ofReal (g t)) s) = fun x =>
-        Ennreal.ofReal (g s) * (Ioi (0 : ℝ)).indicator (fun _ => 1) s *
+      (fun x => (Ioc 0 (f x)).indicator (fun t : ℝ => ENNReal.ofReal (g t)) s) = fun x =>
+        ENNReal.ofReal (g s) * (Ioi (0 : ℝ)).indicator (fun _ => 1) s *
           (Ici s).indicator (fun t : ℝ => (1 : ℝ≥0∞)) (f x) :=
       by
       funext a
@@ -127,7 +127,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     simp_rw [aux₁]
     rw [lintegral_const_mul']
     swap
-    · apply Ennreal.mul_ne_top Ennreal.ofReal_ne_top
+    · apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top
       by_cases s ∈ Ioi (0 : ℝ) <;> · simp [h]
     simp_rw [show
         (fun a => (Ici s).indicator (fun t : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
@@ -144,12 +144,12 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
         (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ { a : α | s ≤ f a } =
           (Ioi 0).indicator (fun _x : ℝ => 1 * μ { a : α | s ≤ f a }) s
         by by_cases 0 < s <;> simp [h]]
-    simp_rw [mul_comm _ (Ennreal.ofReal _), one_mul]
+    simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]
     rfl
   have aux₂ :
     (Function.uncurry fun (x : α) (y : ℝ) =>
-        (Ioc 0 (f x)).indicator (fun t : ℝ => Ennreal.ofReal (g t)) y) =
-      { p : α × ℝ | p.2 ∈ Ioc 0 (f p.1) }.indicator fun p => Ennreal.ofReal (g p.2) :=
+        (Ioc 0 (f x)).indicator (fun t : ℝ => ENNReal.ofReal (g t)) y) =
+      { p : α × ℝ | p.2 ∈ Ioc 0 (f p.1) }.indicator fun p => ENNReal.ofReal (g p.2) :=
     by
     funext p
     cases p
@@ -180,8 +180,8 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, Ennreal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * Ennreal.ofReal (g t) :=
+    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t) :=
   by
   have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G :=
     by
@@ -196,8 +196,8 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrable_on_empty
   have eq₁ :
-    (∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * Ennreal.ofReal (g t)) =
-      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * Ennreal.ofReal (G t) :=
+    (∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (g t)) =
+      ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (G t) :=
     by
     apply lintegral_congr_ae
     filter_upwards [g_eq_G]with a ha
@@ -224,14 +224,14 @@ See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) :
-    (∫⁻ ω, Ennreal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } :=
+    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } :=
   by
   set cst := fun t : ℝ => (1 : ℝ) with def_cst
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ => intervalIntegrableConst
   have key :=
     lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
       (eventually_of_forall fun t => zero_le_one)
-  simp_rw [def_cst, Ennreal.ofReal_one, mul_one] at key
+  simp_rw [def_cst, ENNReal.ofReal_one, mul_one] at key
   rw [← key]
   congr with ω
   simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]
@@ -246,8 +246,8 @@ See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, Ennreal.ofReal (f ω ^ p) ∂μ) =
-      Ennreal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * Ennreal.ofReal (t ^ (p - 1)) :=
+    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t ≤ f a } * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   have one_lt_p : -1 < p - 1 := by linarith
   have obs : ∀ x : ℝ, (∫ t : ℝ in 0 ..x, t ^ (p - 1)) = x ^ p / p :=
@@ -266,10 +266,10 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     intervalIntegral.intervalIntegrableRpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
   simp_rw [g_def] at key
-  rw [← key, ← lintegral_const_mul (Ennreal.ofReal p)] <;> simp_rw [obs]
+  rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
-    rw [← Ennreal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
-  · exact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal
+    rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
+  · exact ((f_mble.pow measurable_const).div_const p).eNNReal_ofReal
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
 
 end MeasureTheory
@@ -344,8 +344,8 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, Ennreal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
-      ∫⁻ t in Ioi 0, μ { a : α | t < f a } * Ennreal.ofReal (g t) :=
+    (∫⁻ ω, ENNReal.ofReal (∫ t in 0 ..f ω, g t) ∂μ) =
+      ∫⁻ t in Ioi 0, μ { a : α | t < f a } * ENNReal.ofReal (g t) :=
   by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
   apply lintegral_congr_ae
@@ -362,7 +362,7 @@ See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) :
-    (∫⁻ ω, Ennreal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t < f a } :=
+    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ { a : α | t < f a } :=
   by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
@@ -379,11 +379,11 @@ See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, Ennreal.ofReal (f ω ^ p) ∂μ) =
-      Ennreal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t < f a } * Ennreal.ofReal (t ^ (p - 1)) :=
+    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ { a : α | t < f a } * ENNReal.ofReal (t ^ (p - 1)) :=
   by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
-  apply congr_arg fun z => Ennreal.ofReal p * z
+  apply congr_arg fun z => ENNReal.ofReal p * z
   apply lintegral_congr_ae
   filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble]with t ht
   rw [ht]

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

Done with a global search and replace, and then (to fix the #align lines), replace (#align \S*)setIntegral with $1set_integral.

@@ -609,7 +609,7 @@ lemma Integrable.integral_eq_integral_Ioc_meas_le {f : α → ℝ} {M : ℝ}
     (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) (f_bdd : f ≤ᵐ[μ] (fun _ ↦ M)) :
     ∫ ω, f ω ∂μ = ∫ t in Ioc 0 M, ENNReal.toReal (μ {a : α | t ≤ f a}) := by
   rw [f_intble.integral_eq_integral_meas_le f_nn]
-  rw [set_integral_eq_of_subset_of_ae_diff_eq_zero
+  rw [setIntegral_eq_of_subset_of_ae_diff_eq_zero
       measurableSet_Ioi.nullMeasurableSet Ioc_subset_Ioi_self ?_]
   apply eventually_of_forall (fun t ht ↦ ?_)
   have htM : M < t := by simp_all only [mem_diff, mem_Ioi, mem_Ioc, not_and, not_le]

A mix of various changes; generated with a script and manually tweaked.

@@ -37,18 +37,21 @@ are also included.
 
 ## Main results
 
- * `lintegral_comp_eq_lintegral_meas_le_mul` and `lintegral_comp_eq_lintegral_meas_lt_mul`:
+ * `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul`
+   and `MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul`:
    The general layer cake formulas with Lebesgue integrals, written in terms of measures of
    sets of the forms {ω | t ≤ f(ω)} and {ω | t < f(ω)}, respectively.
- * `lintegral_eq_lintegral_meas_le` and `lintegral_eq_lintegral_meas_lt`:
+ * `MeasureTheory.lintegral_eq_lintegral_meas_le` and
+   `MeasureTheory.lintegral_eq_lintegral_meas_lt`:
    The most common special cases of the layer cake formulas, stating that for a nonnegative
    function f we have ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) ≥ t} dt and
    ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) > t} dt, respectively.
- * `lintegral_rpow_eq_lintegral_meas_le_mul` and `lintegral_rpow_eq_lintegral_meas_lt_mul`:
+ * `MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul` and
+   `MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul`:
    Other common special cases of the layer cake formulas, stating that for a nonnegative function f
    and p > 0, we have ∫ f(ω)^p ∂μ(ω) = p * ∫ μ {ω | f(ω) ≥ t} * t^(p-1) dt and
    ∫ f(ω)^p ∂μ(ω) = p * ∫ μ {ω | f(ω) > t} * t^(p-1) dt, respectively.
- * `integral_eq_integral_meas_lt`:
+ * `Integrable.integral_eq_integral_meas_lt`:
    A Bochner integral version of the most common special case of the layer cake formulas, stating
    that for an integrable and a.e.-nonnegative function f we have
    ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) > t} dt.
@@ -100,7 +103,7 @@ formula), with a measurability assumption that would also essentially follow fro
 integrability assumptions, and a sigma-finiteness assumption.
 
 See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
-`lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
+`MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
 cake formula. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     (μ : Measure α) [SigmaFinite μ]
@@ -190,7 +193,7 @@ Compared to `lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinit
 the sigma-finite assumption.
 
 See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
-`lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
+`MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
 cake formula. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     (f_nn : 0 ≤ f) (f_mble : Measurable f)
@@ -390,8 +393,8 @@ weighted by `g`.
 
 Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞, g(t) * μ {ω | f(ω) ≥ t}`.
 
-See `lintegral_comp_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
-instead. -/
+See `MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul` for a version with sets of the form
+`{ω | f(ω) > t}` instead. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
@@ -442,8 +445,8 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤
 For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0..∞, μ {ω | f(ω) ≥ t}`.
 
-See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω | f(ω) > t}`
-instead. -/
+See `MeasureTheory.lintegral_eq_lintegral_meas_lt` for a version with sets of the form
+`{ω | f(ω) > t}` instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) :
     ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
@@ -464,8 +467,8 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
 For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) ≥ t}`.
 
-See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
-instead. -/
+See `MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the form
+`{ω | f(ω) > t}` instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
     ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
@@ -571,7 +574,8 @@ variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {f : α → ℝ}
 For an integrable a.e.-nonnegative real-valued function `f`, the Bochner integral of `f` can be
 written (roughly speaking) as: `∫ f ∂μ = ∫ t in 0..∞, μ {ω | f(ω) > t}`.
 
-See `lintegral_eq_lintegral_meas_lt` for a version with Lebesgue integral `∫⁻` instead. -/
+See `MeasureTheory.lintegral_eq_lintegral_meas_lt` for a version with Lebesgue integral `∫⁻`
+instead. -/
 theorem Integrable.integral_eq_integral_meas_lt
     (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
     ∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

| Statement | New name | Old name | |

@@ -485,7 +485,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
   rw [← key, ← lintegral_const_mul'' (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
-    rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
+    rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel₀ (f ω ^ p) p_pos.ne.symm]
   · have aux := (@measurable_const ℝ α (by infer_instance) (by infer_instance) p).aemeasurable
                   (μ := μ)
     exact (Measurable.ennreal_ofReal (hf := measurable_id)).comp_aemeasurable

Empty lines were removed by executing the following Python script twice

import os
import re


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

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

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

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

@@ -497,9 +497,7 @@ end Layercake
 section LayercakeLT
 
 variable {α : Type*} [MeasurableSpace α] (μ : Measure α)
-
 variable {β : Type*} [MeasurableSpace β] [MeasurableSingletonClass β]
-
 variable {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:

This PR establishes an easy topology comparison: the topology given by the Lévy-Prokhorov distance is finer than the topology of convergence in distribution.

Co-authored-by: kkytola <39528102+kkytola@users.noreply.github.com> Co-authored-by: kkytola <“kalle.kytola@aalto.fi”> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -107,7 +107,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     (f_nn : 0 ≤ f) (f_mble : Measurable f)
     (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
     (g_nn : ∀ t > 0, 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
   have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t := by
     intro t ht
@@ -115,7 +115,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     · simp [← h]
     · exact g_intble t h
   have integrand_eq : ∀ ω,
-      ENNReal.ofReal (∫ t in (0)..f ω, g t) = (∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t)) := by
+      ENNReal.ofReal (∫ t in (0)..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) := by
     intro ω
     have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
       filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))]
@@ -196,7 +196,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     (f_nn : 0 ≤ f) (f_mble : Measurable f)
     (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
     (g_nn : ∀ t > 0, 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
   /- We will reduce to the sigma-finite case, after excluding two easy cases where the result
   is more or less obvious. -/
@@ -395,7 +395,7 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
   obtain ⟨G, G_mble, G_nn, g_eq_G⟩ : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G
       ∧ g =ᵐ[volume.restrict (Ioi 0)] G := by
@@ -414,7 +414,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤
     rw [← hω]
     exact (max_eq_left h'ω).symm
   have eq₁ :
-    (∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)) =
+    ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ F a} * ENNReal.ofReal (G t) := by
     apply lintegral_congr_ae
     filter_upwards [g_eq_G] with t ht
@@ -446,7 +446,7 @@ See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
+    ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
   set cst := fun _ : ℝ => (1 : ℝ)
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
     intervalIntegrable_const
@@ -468,10 +468,10 @@ See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by
   have one_lt_p : -1 < p - 1 := by linarith
-  have obs : ∀ x : ℝ, (∫ t : ℝ in (0)..x, t ^ (p - 1)) = x ^ p / p := by
+  have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
     intro x
     rw [integral_rpow (Or.inl one_lt_p)]
     simp [Real.zero_rpow p_pos.ne.symm]
@@ -517,7 +517,7 @@ instead. -/
 theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
-    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
       ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) := by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
   apply lintegral_congr_ae
@@ -535,7 +535,7 @@ See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω |
 instead. -/
 theorem lintegral_eq_lintegral_meas_lt (μ : Measure α)
     (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) :
-    (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t < f a} := by
+    ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t < f a} := by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
   filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
@@ -552,7 +552,7 @@ See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the for
 instead. -/
 theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α)
     (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
-    (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
+    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) := by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
   apply congr_arg fun z => ENNReal.ofReal p * z
@@ -576,7 +576,7 @@ written (roughly speaking) as: `∫ f ∂μ = ∫ t in 0..∞, μ {ω | f(ω) >
 See `lintegral_eq_lintegral_meas_lt` for a version with Lebesgue integral `∫⁻` instead. -/
 theorem Integrable.integral_eq_integral_meas_lt
     (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
-    (∫ ω, f ω ∂μ) = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by
+    ∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by
   have key := lintegral_eq_lintegral_meas_lt μ f_nn f_intble.aemeasurable
   have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞ := Integrable.lintegral_lt_top f_intble
   have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} < ∞ := by simp only [← key, lhs_finite]
@@ -595,6 +595,28 @@ theorem Integrable.integral_eq_integral_meas_lt
       refine Measurable.ennreal_toReal ?_
       exact Antitone.measurable (fun _ _ hst ↦ measure_mono (fun _ h ↦ lt_of_le_of_lt hst h))
 
+theorem Integrable.integral_eq_integral_meas_le
+    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
+    ∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t ≤ f a}) := by
+  rw [Integrable.integral_eq_integral_meas_lt f_intble f_nn]
+  apply integral_congr_ae
+  filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f] with t ht
+  exact congrArg ENNReal.toReal ht.symm
+
+lemma Integrable.integral_eq_integral_Ioc_meas_le {f : α → ℝ} {M : ℝ}
+    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) (f_bdd : f ≤ᵐ[μ] (fun _ ↦ M)) :
+    ∫ ω, f ω ∂μ = ∫ t in Ioc 0 M, ENNReal.toReal (μ {a : α | t ≤ f a}) := by
+  rw [f_intble.integral_eq_integral_meas_le f_nn]
+  rw [set_integral_eq_of_subset_of_ae_diff_eq_zero
+      measurableSet_Ioi.nullMeasurableSet Ioc_subset_Ioi_self ?_]
+  apply eventually_of_forall (fun t ht ↦ ?_)
+  have htM : M < t := by simp_all only [mem_diff, mem_Ioi, mem_Ioc, not_and, not_le]
+  have obs : μ {a | M < f a} = 0 := by
+    rw [measure_zero_iff_ae_nmem]
+    filter_upwards [f_bdd] with a ha using not_lt.mpr ha
+  rw [ENNReal.toReal_eq_zero_iff]
+  exact Or.inl <| measure_mono_null (fun a ha ↦ lt_of_lt_of_le htM ha) obs
+
 end LayercakeIntegral
 
 end MeasureTheory

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -453,7 +453,7 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
   have key :=
     lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
       (eventually_of_forall fun _ => zero_le_one)
-  simp_rw [ENNReal.ofReal_one, mul_one] at key
+  simp_rw [cst, ENNReal.ofReal_one, mul_one] at key
   rw [← key]
   congr with ω
   simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]

Redefine ≤ on MeasureTheory.Measure so that μ ≤ ν ↔ ∀ s, μ s ≤ ν s by definition instead of ∀ s, MeasurableSet s → μ s ≤ ν s.

Reasons

• this way it is defeq to ≤ on outer measures;
• if we decide to introduce an order on all DFunLike types and migrate measures to FunLike, then this is unavoidable;
• the reasoning for the old definition was "it's slightly easier to prove μ ≤ ν this way"; the counter-argument is "it's slightly harder to apply μ ≤ ν this way".

Other changes

• golf some proofs broken by this change;
• add @[gcongr] tags to some ENNReal lemmas;
• fix the name ENNReal.coe_lt_coe_of_le -> ENNReal.ENNReal.coe_lt_coe_of_lt;
• drop an unneeded MeasurableSet assumption in set_lintegral_pdf_le_map

@@ -331,7 +331,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
         refine lt_of_le_of_lt (measure_union_le _ _) ?_
         rw [I, zero_add]
         apply lt_of_le_of_lt _ J
-        exact restrict_le_self _ (measurableSet_lt measurable_const f_mble)
+        exact restrict_le_self _
       spanning := by
         apply eq_univ_iff_forall.2 (fun a ↦ ?_)
         rcases le_or_lt (f a) M with ha|ha

This better matches other lemma names.

From LeanAPAP

@@ -479,7 +479,7 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤
   have g_nn : ∀ᵐ t ∂volume.restrict (Ioi (0 : ℝ)), 0 ≤ g t := by
     filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : ℝ)))]
     intro t t_pos
-    exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)
+    exact Real.rpow_nonneg (mem_Ioi.mp t_pos).le (p - 1)
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
     intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn

@@ -380,7 +380,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
   exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     ν f_nn f_mble g_intble g_mble g_nn
 
-/-- The layer cake formula / Cavalieri's principle / tail probability formula:
+/-- The layer cake formula / **Cavalieri's principle** / tail probability formula:
 
 Let `f` be a non-negative measurable function on a measure space. Let `G` be an
 increasing absolutely continuous function on the positive real line, vanishing at the origin,

I've also got a change to make this required, but I'd like to land this first.

@@ -134,7 +134,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
         ENNReal.ofReal (g s) * (Ioi (0 : ℝ)).indicator (fun _ => 1) s *
           (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f x) := by
       funext a
-      by_cases s ∈ Ioc (0 : ℝ) (f a)
+      by_cases h : s ∈ Ioc (0 : ℝ) (f a)
       · simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
           mul_one]
       · have h_copy := h
@@ -152,7 +152,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     simp_rw [show
         (fun a => (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
           {a : α | s ≤ f a}.indicator (fun _ => 1) a
-        by funext a; by_cases s ≤ f a <;> simp [h]]
+        by funext a; by_cases h : s ≤ f a <;> simp [h]]
     rw [lintegral_indicator₀]
     swap; · exact f_mble.nullMeasurable measurableSet_Ici
     rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,
@@ -160,7 +160,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
       show
         (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ {a : α | s ≤ f a} =
           (Ioi 0).indicator (fun _x : ℝ => 1 * μ {a : α | s ≤ f a}) s
-        by by_cases 0 < s <;> simp [h]]
+        by by_cases h : 0 < s <;> simp [h]]
     simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]
     rfl
   have aux₂ :
@@ -170,7 +170,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
     funext p
     cases p with | mk p_fst p_snd => ?_
     rw [Function.uncurry_apply_pair]
-    by_cases p_snd ∈ Ioc 0 (f p_fst)
+    by_cases h : p_snd ∈ Ioc 0 (f p_fst)
     · have h' : (p_fst, p_snd) ∈ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h
       rw [Set.indicator_of_mem h', Set.indicator_of_mem h]
     · have h' : (p_fst, p_snd) ∉ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h

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

@@ -79,7 +79,7 @@ theorem countable_meas_le_ne_meas_lt (g : α → R) :
   intro t ht
   have : μ {a | t < g a} < μ {a | t ≤ g a} :=
     lt_of_le_of_ne (measure_mono (fun a ha ↦ le_of_lt ha)) (Ne.symm ht)
-  refine ⟨μ {a | t < g a}, this, fun s hs ↦ measure_mono (fun a ha ↦ hs.trans_le ha)⟩
+  exact ⟨μ {a | t < g a}, this, fun s hs ↦ measure_mono (fun a ha ↦ hs.trans_le ha)⟩
 
 theorem meas_le_ae_eq_meas_lt {R : Type*} [LinearOrder R] [MeasurableSpace R]
     (ν : Measure R) [NoAtoms ν] (g : α → R) :

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

@@ -178,7 +178,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
   rw [aux₂]
   have mble₀ : MeasurableSet {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := by
     simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using
-      measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ
+      measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
   exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator₀
     mble₀.nullMeasurableSet
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite

Tag lemmas about tsub (truncated subtraction), nnreal and ennreal powers, and measures for gcongr.

@@ -237,7 +237,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
           apply set_lintegral_mono' measurableSet_Ioc (fun x hx ↦ ?_)
           rw [← h's]
           gcongr
-          exact measure_mono (fun a ha ↦ hx.2.trans (le_of_lt ha))
+          exact fun a ha ↦ hx.2.trans (le_of_lt ha)
       _ ≤ ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
           lintegral_mono_set Ioc_subset_Ioi_self
     /- The second integral is infinite, as one integrates amont other things on those `ω` where

Currently, the layercake formula for the Lebesgue integral assumes sigma-finiteness of the measure, while the layercake formula for the Bochner integral (and integrable functions) doesn't. At the cost of a more complicated proof, we remove the sigma-finiteness also from the Lebesgue measure case.

Co-authored-by: Kalle <kalle.kytola@aalto.fi>

@@ -5,7 +5,6 @@ Authors: Kalle Kytölä
 -/
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
 import Mathlib.Analysis.SpecialFunctions.Integrals
-import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
 
 #align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
 
@@ -14,7 +13,7 @@ import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
 
 In this file we prove the following layer cake formula.
 
-Consider a non-negative measurable function `f` on a sigma-finite measure space. Apply pointwise
+Consider a non-negative measurable function `f` on a measure space. Apply pointwise
 to it an increasing absolutely continuous function `G : ℝ≥0 → ℝ≥0` vanishing at the origin, with
 derivative `G' = g` on the positive real line (in other words, `G` a primitive of a non-negative
 locally integrable function `g` on the positive real line). Then the integral of the result,
@@ -52,40 +51,60 @@ are also included.
  * `integral_eq_integral_meas_lt`:
    A Bochner integral version of the most common special case of the layer cake formulas, stating
    that for an integrable and a.e.-nonnegative function f we have
-   ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) > t} dt. In this result, sigma-finiteness of μ does not need to be
-   explicitly assumed, because integrability guarantees sigma-finiteness of the restriction of μ
-   to the support of f.
+   ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) > t} dt.
 
 ## Tags
 
 layer cake representation, Cavalieri's principle, tail probability formula
 -/
 
-
 noncomputable section
 
-open scoped ENNReal MeasureTheory
+open scoped ENNReal MeasureTheory Topology
+
+open Set MeasureTheory Filter Measure
+
+namespace MeasureTheory
+
+section
 
-open Set MeasureTheory Filter
+variable {α R : Type*} [MeasurableSpace α] (μ : Measure α) [LinearOrder R]
+
+theorem countable_meas_le_ne_meas_lt (g : α → R) :
+    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
+  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
+  -- jumps down on the right of `t`. This jump set is countable for any function.
+  let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
+  apply (countable_image_gt_image_Ioi F).mono
+  intro t ht
+  have : μ {a | t < g a} < μ {a | t ≤ g a} :=
+    lt_of_le_of_ne (measure_mono (fun a ha ↦ le_of_lt ha)) (Ne.symm ht)
+  refine ⟨μ {a | t < g a}, this, fun s hs ↦ measure_mono (fun a ha ↦ hs.trans_le ha)⟩
+
+theorem meas_le_ae_eq_meas_lt {R : Type*} [LinearOrder R] [MeasurableSpace R]
+    (ν : Measure R) [NoAtoms ν] (g : α → R) :
+    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
+  Set.Countable.measure_zero (countable_meas_le_ne_meas_lt μ g) _
+
+end
 
 /-! ### Layercake formula -/
 
 
 section Layercake
 
-namespace MeasureTheory
-
 variable {α : Type*} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
 /-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
 formula), with a measurability assumption that would also essentially follow from the
-integrability assumptions.
+integrability assumptions, and a sigma-finiteness assumption.
 
 See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
 `lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
 cake formula. -/
-theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α) [SigmaFinite μ]
-    (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ)
+theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
+    (μ : Measure α) [SigmaFinite μ]
+    (f_nn : 0 ≤ f) (f_mble : Measurable f)
     (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
     (g_nn : ∀ t > 0, 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
@@ -95,16 +114,16 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     cases' eq_or_lt_of_le ht with h h
     · simp [← h]
     · exact g_intble t h
-  have integrand_eq : ∀ᵐ ω ∂μ,
+  have integrand_eq : ∀ ω,
       ENNReal.ofReal (∫ t in (0)..f ω, g t) = (∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t)) := by
-    filter_upwards [f_nn] with ω fω_nn
+    intro ω
     have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
       filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))]
         with x hx using g_nn x hx.1
-    rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) fω_nn).1 g_ae_nn]
+    rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]
     congr
-    exact intervalIntegral.integral_of_le fω_nn
-  rw [lintegral_congr_ae integrand_eq]
+    exact intervalIntegral.integral_of_le (f_nn ω)
+  rw [lintegral_congr integrand_eq]
   simp_rw [← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]
   -- Porting note: was part of `simp_rw` on the previous line, but didn't trigger.
   rw [← lintegral_indicator _ measurableSet_Ioi, lintegral_lintegral_swap]
@@ -157,15 +176,213 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     · have h' : (p_fst, p_snd) ∉ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h
       rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]
   rw [aux₂]
-  have mble₀ : NullMeasurableSet {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} (μ.prod volume) := by
+  have mble₀ : MeasurableSet {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := by
     simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using
-      nullMeasurableSet_region_between_oc μ measurable_zero.aemeasurable f_mble MeasurableSet.univ
-  exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator₀ mble₀
-#align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
+      measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ
+  exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator₀
+    mble₀.nullMeasurableSet
+#align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
+
+/-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
+formula), with a measurability assumption that would also essentially follow from the
+integrability assumptions.
+Compared to `lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite`, we remove
+the sigma-finite assumption.
+
+See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
+`lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
+cake formula. -/
+theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
+    (f_nn : 0 ≤ f) (f_mble : Measurable f)
+    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
+    (g_nn : ∀ t > 0, 0 ≤ g t) :
+    (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
+      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
+  /- We will reduce to the sigma-finite case, after excluding two easy cases where the result
+  is more or less obvious. -/
+  have f_nonneg : ∀ ω, 0 ≤ f ω := fun ω ↦ f_nn ω
+  -- trivial case where `g` is ae zero. Then both integrals vanish.
+  by_cases H1 : g =ᵐ[volume.restrict (Ioi (0 : ℝ))] 0
+  · have A : ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ = 0 := by
+      have : ∀ ω, ∫ t in (0)..f ω, g t = ∫ t in (0)..f ω, 0 := by
+        intro ω
+        simp_rw [intervalIntegral.integral_of_le (f_nonneg ω)]
+        apply integral_congr_ae
+        exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1
+      simp [this]
+    have B : ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) = 0 := by
+      have : (fun t ↦ μ {a : α | t ≤ f a} * ENNReal.ofReal (g t))
+        =ᵐ[volume.restrict (Ioi (0:ℝ))] 0 := by
+          filter_upwards [H1] with t ht using by simp [ht]
+      simp [lintegral_congr_ae this]
+    rw [A, B]
+  -- easy case where both sides are obviously infinite: for some `s`, one has
+  -- `μ {a : α | s < f a} = ∞` and moreover `g` is not ae zero on `[0, s]`.
+  by_cases H2 : ∃ s > 0, 0 < ∫ t in (0)..s, g t ∧ μ {a : α | s < f a} = ∞
+  · rcases H2 with ⟨s, s_pos, hs, h's⟩
+    rw [intervalIntegral.integral_of_le s_pos.le] at hs
+    /- The first integral is infinite, as for `t ∈ [0, s]` one has `μ {a : α | t ≤ f a} = ∞`,
+    and moreover the additional integral `g` is not uniformly zero. -/
+    have A : ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) = ∞ := by
+      rw [eq_top_iff]
+      calc
+      ∞ = ∫⁻ t in Ioc 0 s, ∞ * ENNReal.ofReal (g t) := by
+          have I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0 := by
+            rw [← ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1]
+            · simpa only [not_lt, ge_iff_le, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs
+            · filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1
+          rw [lintegral_const_mul, ENNReal.top_mul I_pos]
+          exact ENNReal.measurable_ofReal.comp g_mble
+      _ ≤ ∫⁻ t in Ioc 0 s, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
+          apply set_lintegral_mono' measurableSet_Ioc (fun x hx ↦ ?_)
+          rw [← h's]
+          gcongr
+          exact measure_mono (fun a ha ↦ hx.2.trans (le_of_lt ha))
+      _ ≤ ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) :=
+          lintegral_mono_set Ioc_subset_Ioi_self
+    /- The second integral is infinite, as one integrates amont other things on those `ω` where
+    `f ω > s`: this is an infinite measure set, and on it the integrand is bounded below
+    by `∫ t in 0..s, g t` which is positive. -/
+    have B : ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ = ∞ := by
+      rw [eq_top_iff]
+      calc
+      ∞ = ∫⁻ _ in {a | s < f a}, ENNReal.ofReal (∫ t in (0)..s, g t) ∂μ := by
+          simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter,
+            h's, ne_eq, ENNReal.ofReal_eq_zero, not_le]
+          rw [ENNReal.mul_top]
+          simpa [intervalIntegral.integral_of_le s_pos.le] using hs
+      _ ≤ ∫⁻ ω in {a | s < f a}, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ := by
+          apply set_lintegral_mono' (measurableSet_lt measurable_const f_mble) (fun a ha ↦ ?_)
+          apply ENNReal.ofReal_le_ofReal
+          apply intervalIntegral.integral_mono_interval le_rfl s_pos.le (le_of_lt ha)
+          · filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1
+          · exact g_intble _ (s_pos.trans ha)
+      _ ≤ ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ := set_lintegral_le_lintegral _ _
+    rw [A, B]
+  /- It remains to handle the interesting case, where `g` is not zero, but both integrals are
+  not obviously infinite. Let `M` be the largest number such that `g = 0` on `[0, M]`. Then we
+  may restrict `μ` to the points where `f ω > M` (as the other ones do not contribute to the
+  integral). The restricted measure `ν` is sigma-finite, as `μ` gives finite measure to
+  `{ω | f ω > a}` for any `a > M` (otherwise, we would be in the easy case above), so that
+  one can write (a full measure subset of) the space as the countable union of the finite measure
+  sets `{ω | f ω > uₙ}` for `uₙ` a sequence decreasing to `M`. Therefore,
+  this case follows from the case where the measure is sigma-finite, applied to `ν`. -/
+  push_neg at H2
+  have M_bdd : BddAbove {s : ℝ | g =ᵐ[volume.restrict (Ioc (0 : ℝ) s)] 0} := by
+    contrapose! H1
+    have : ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc (0 : ℝ) n)] 0 := by
+      intro n
+      rcases not_bddAbove_iff.1 H1 n with ⟨s, hs, ns⟩
+      exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs
+    have Hg : g =ᵐ[volume.restrict (⋃ (n : ℕ), (Ioc (0 : ℝ) n))] 0 :=
+      (ae_restrict_iUnion_iff _ _).2 this
+    have : (⋃ (n : ℕ), (Ioc (0 : ℝ) n)) = Ioi 0 :=
+      iUnion_Ioc_eq_Ioi_self_iff.2 (fun x _ ↦ exists_nat_ge x)
+    rwa [this] at Hg
+  -- let `M` be the largest number such that `g` vanishes ae on `(0, M]`.
+  let M : ℝ := sSup {s : ℝ | g =ᵐ[volume.restrict (Ioc (0 : ℝ) s)] 0}
+  have zero_mem : 0 ∈ {s : ℝ | g =ᵐ[volume.restrict (Ioc (0 : ℝ) s)] 0} := by simpa using trivial
+  have M_nonneg : 0 ≤ M := le_csSup M_bdd zero_mem
+  -- Then the function `g` indeed vanishes ae on `(0, M]`.
+  have hgM : g =ᵐ[volume.restrict (Ioc (0 : ℝ) M)] 0 := by
+    rw [← restrict_Ioo_eq_restrict_Ioc]
+    obtain ⟨u, -, uM, ulim⟩ : ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < M) ∧ Tendsto u atTop (𝓝 M) :=
+      exists_seq_strictMono_tendsto M
+    have I : ∀ n, g =ᵐ[volume.restrict (Ioc (0 : ℝ) (u n))] 0 := by
+      intro n
+      obtain ⟨s, hs, uns⟩ : ∃ s, g =ᶠ[ae (Measure.restrict volume (Ioc 0 s))] 0 ∧ u n < s :=
+        exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)
+      exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs
+    have : g =ᵐ[volume.restrict (⋃ n, Ioc (0 : ℝ) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I
+    apply ae_restrict_of_ae_restrict_of_subset _ this
+    rintro x ⟨x_pos, xM⟩
+    obtain ⟨n, hn⟩ : ∃ n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists
+    exact mem_iUnion.2 ⟨n, ⟨x_pos, hn.le⟩⟩
+  -- Let `ν` be the restriction of `μ` to those points where `f a > M`.
+  let ν := μ.restrict {a : α | M < f a}
+  -- This measure is sigma-finite (this is the whole point of the argument).
+  have : SigmaFinite ν := by
+    obtain ⟨u, -, uM, ulim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), M < u n) ∧ Tendsto u atTop (𝓝 M) :=
+      exists_seq_strictAnti_tendsto M
+    let s : ν.FiniteSpanningSetsIn univ :=
+    { set := fun n ↦ {a | f a ≤ M} ∪ {a | u n < f a}
+      set_mem := fun _ ↦ trivial
+      finite := by
+        intro n
+        have I : ν {a | f a ≤ M} = 0 := by
+          rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]
+          convert measure_empty
+          rw [← disjoint_iff_inter_eq_empty]
+          exact disjoint_left.mpr (fun a ha ↦ by simpa using ha)
+        have J : μ {a | u n < f a} < ∞ := by
+          rw [lt_top_iff_ne_top]
+          apply H2 _ (M_nonneg.trans_lt (uM n))
+          by_contra H3
+          rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3
+          have g_nn_ae : ∀ᵐ t ∂(volume.restrict (Ioc 0 (u n))), 0 ≤ g t := by
+            filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1
+          have Ig : ∫ (t : ℝ) in Ioc 0 (u n), g t = 0 :=
+            le_antisymm H3 (integral_nonneg_of_ae g_nn_ae)
+          have J : ∀ᵐ t ∂(volume.restrict (Ioc 0 (u n))), g t = 0 :=
+            (integral_eq_zero_iff_of_nonneg_ae g_nn_ae
+              (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig
+          have : u n ≤ M := le_csSup M_bdd J
+          exact lt_irrefl _ (this.trans_lt (uM n))
+        refine lt_of_le_of_lt (measure_union_le _ _) ?_
+        rw [I, zero_add]
+        apply lt_of_le_of_lt _ J
+        exact restrict_le_self _ (measurableSet_lt measurable_const f_mble)
+      spanning := by
+        apply eq_univ_iff_forall.2 (fun a ↦ ?_)
+        rcases le_or_lt (f a) M with ha|ha
+        · exact mem_iUnion.2 ⟨0, Or.inl ha⟩
+        · obtain ⟨n, hn⟩ : ∃ n, u n < f a := ((tendsto_order.1 ulim).2 _ ha).exists
+          exact mem_iUnion.2 ⟨n, Or.inr hn⟩ }
+    exact ⟨⟨s⟩⟩
+  -- the first integrals with respect to `μ` and to `ν` coincide, as points with `f a ≤ M` are
+  -- weighted by zero as `g` vanishes there.
+  have A : ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ
+         = ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂ν := by
+    have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble
+    have I : ∫⁻ ω in {a | M < f a}ᶜ, ENNReal.ofReal (∫ t in (0).. f ω, g t) ∂μ
+             = ∫⁻ _ in {a | M < f a}ᶜ, 0 ∂μ := by
+      apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs ↦ ?_))
+      have : ∫ (t : ℝ) in (0)..f s, g t = ∫ (t : ℝ) in (0)..f s, 0 := by
+        simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]
+        apply integral_congr_ae
+        apply ae_mono (restrict_mono ?_ le_rfl) hgM
+        apply Ioc_subset_Ioc_right
+        simpa using hs
+      simp [this]
+    simp only [lintegral_const, zero_mul] at I
+    rw [← lintegral_add_compl _ meas, I, add_zero]
+  -- the second integrals with respect to `μ` and to `ν` coincide, as points with `f a ≤ M` do not
+  -- contribute to either integral since the weight `g` vanishes.
+  have B : ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
+           = ∫⁻ t in Ioi 0, ν {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
+    have B1 : ∫⁻ t in Ioc 0 M, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
+         = ∫⁻ t in Ioc 0 M, ν {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
+      apply lintegral_congr_ae
+      filter_upwards [hgM] with t ht
+      simp [ht]
+    have B2 : ∫⁻ t in Ioi M, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
+              = ∫⁻ t in Ioi M, ν {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
+      apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht ↦ ?_))
+      rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]
+      congr 3
+      exact (inter_eq_left.2 (fun a ha ↦ (mem_Ioi.1 ht).trans_le ha)).symm
+    have I : Ioi (0 : ℝ) = Ioc (0 : ℝ) M ∪ Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm
+    have J : Disjoint (Ioc 0 M) (Ioi M) := Ioc_disjoint_Ioi le_rfl
+    rw [I, lintegral_union measurableSet_Ioi J, lintegral_union measurableSet_Ioi J, B1, B2]
+  -- therefore, we may replace the integrals wrt `μ` with integrals wrt `ν`, and apply the
+  -- result for sigma-finite measures.
+  rw [A, B]
+  exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
+    ν f_nn f_mble g_intble g_mble g_nn
 
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
 
-Let `f` be a non-negative measurable function on a sigma-finite measure space. Let `G` be an
+Let `f` be a non-negative measurable function on a measure space. Let `G` be an
 increasing absolutely continuous function on the positive real line, vanishing at the origin,
 with derivative `G' = g`. Then the integral of the composition `G ∘ f` can be written as
 the integral over the positive real line of the "tail measures" `μ {ω | f(ω) ≥ t}` of `f`
@@ -175,49 +392,59 @@ Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞
 
 See `lintegral_comp_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
-  have ex_G : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G ∧ g =ᵐ[volume.restrict (Ioi 0)] G := by
+  obtain ⟨G, G_mble, G_nn, g_eq_G⟩ : ∃ G : ℝ → ℝ, Measurable G ∧ 0 ≤ G
+      ∧ g =ᵐ[volume.restrict (Ioi 0)] G := by
     refine' AEMeasurable.exists_measurable_nonneg _ g_nn
     exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable
-  rcases ex_G with ⟨G, G_mble, G_nn, g_eq_G⟩
   have g_eq_G_on : ∀ t, g =ᵐ[volume.restrict (Ioc 0 t)] G := fun t =>
     ae_mono (Measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G
   have G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t := by
     refine' fun t t_pos => ⟨(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _⟩
     rw [Ioc_eq_empty_of_le t_pos.lt.le]
     exact integrableOn_empty
+  obtain ⟨F, F_mble, F_nn, f_eq_F⟩ : ∃ F : α → ℝ, Measurable F ∧ 0 ≤ F ∧ f =ᵐ[μ] F := by
+    refine ⟨fun ω ↦ max (f_mble.mk f ω) 0, f_mble.measurable_mk.max measurable_const,
+        fun ω ↦ le_max_right _ _, ?_⟩
+    filter_upwards [f_mble.ae_eq_mk, f_nn] with ω hω h'ω
+    rw [← hω]
+    exact (max_eq_left h'ω).symm
   have eq₁ :
     (∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)) =
-      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (G t) := by
+      ∫⁻ t in Ioi 0, μ {a : α | t ≤ F a} * ENNReal.ofReal (G t) := by
     apply lintegral_congr_ae
-    filter_upwards [g_eq_G] with a ha
-    rw [ha]
+    filter_upwards [g_eq_G] with t ht
+    rw [ht]
+    congr 1
+    apply measure_congr
+    filter_upwards [f_eq_F] with a ha using by simp [setOf, ha]
   have eq₂ : ∀ᵐ ω ∂μ,
-      ENNReal.ofReal (∫ t in (0)..f ω, g t) = ENNReal.ofReal (∫ t in (0)..f ω, G t) := by
-    filter_upwards [f_nn] with ω fω_nn
+      ENNReal.ofReal (∫ t in (0)..f ω, g t) = ENNReal.ofReal (∫ t in (0)..F ω, G t) := by
+    filter_upwards [f_eq_F] with ω fω_nn
+    rw [fω_nn]
     congr 1
     refine' intervalIntegral.integral_congr_ae _
-    have fω_nn : 0 ≤ f ω := fω_nn
+    have fω_nn : 0 ≤ F ω := F_nn ω
     rw [uIoc_of_le fω_nn, ←
-      ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : ℝ) (f ω)))]
-    exact g_eq_G_on (f ω)
+      ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : ℝ) (F ω)))]
+    exact g_eq_G_on (F ω)
   simp_rw [lintegral_congr_ae eq₂, eq₁]
-  exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable μ f_nn f_mble
+  exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable μ F_nn F_mble
           G_intble G_mble (fun t _ => G_nn t)
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul
 
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
-For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
+For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0..∞, μ {ω | f(ω) ≥ t}`.
 
 See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) :
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
   set cst := fun _ : ℝ => (1 : ℝ)
@@ -234,12 +461,12 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
 
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
-For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
+For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) ≥ t}`.
 
 See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by
@@ -265,67 +492,19 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
       ((f_mble.pow aux).div_const p)
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
 
-end MeasureTheory
-
 end Layercake
 
 section LayercakeLT
 
-open MeasureTheory
-
 variable {α : Type*} [MeasurableSpace α] (μ : Measure α)
 
 variable {β : Type*} [MeasurableSpace β] [MeasurableSingletonClass β]
 
-namespace Measure
-
-theorem meas_eq_pos_of_meas_le_ne_meas_lt
-    {α : Type*} [MeasurableSpace α] {μ : Measure α} {R : Type*} [LinearOrder R]
-    {g : α → R} {t : R} (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) :
-    0 < μ {a : α | g a = t} := by
-  by_contra con
-  rw [not_lt, nonpos_iff_eq_zero] at con
-  apply ht
-  refine le_antisymm ?_ (measure_mono (fun a ha ↦ le_of_lt ha))
-  have uni : {a : α | t ≤ g a} = {a : α | t < g a} ∪ {a : α | t = g a} := by
-    ext a
-    simpa only [mem_setOf, mem_union] using le_iff_lt_or_eq
-  rw [show {a : α | t = g a} = {a : α | g a = t} by simp_rw [eq_comm]] at uni
-  have μ_le_add := measure_union_le (μ := μ) {a : α | t < g a} {a : α | g a = t}
-  rwa [con, add_zero, ← uni] at μ_le_add
-#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_eq_pos_of_meas_le_ne_meas_lt
-
-theorem countable_meas_le_ne_meas_lt₀ [SigmaFinite μ] {R : Type*} [LinearOrder R]
-    [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : NullMeasurable g μ) :
-    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
-  Countable.mono (fun _ h ↦ meas_eq_pos_of_meas_le_ne_meas_lt h)
-    (Measure.countable_meas_level_set_pos₀ g_mble)
-
-theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R]
-    [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
-    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
-  countable_meas_le_ne_meas_lt₀ (μ := μ) g_mble.nullMeasurable
-#align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
-
-theorem meas_le_ae_eq_meas_lt₀ [SigmaFinite μ] {R : Type*} [LinearOrder R] [MeasurableSpace R]
-    [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R}
-    (g_mble : NullMeasurable g μ) :
-    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
-  Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt₀ μ g_mble) _
-
-theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R] [MeasurableSpace R]
-    [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
-    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
-  Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _
-#align measure.meas_le_ae_eq_meas_lt Measure.meas_le_ae_eq_meas_lt
-
-end Measure
-
 variable {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
 
-Let `f` be a non-negative measurable function on a sigma-finite measure space. Let `G` be an
+Let `f` be a non-negative measurable function on a measure space. Let `G` be an
 increasing absolutely continuous function on the positive real line, vanishing at the origin,
 with derivative `G' = g`. Then the integral of the composition `G ∘ f` can be written as
 the integral over the positive real line of the "tail measures" `μ {ω | f(ω) > t}` of `f`
@@ -335,60 +514,58 @@ Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞
 
 See `lintegral_comp_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
     (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
       ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) := by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+  filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
     with t ht
   rw [ht]
-#align lintegral_comp_eq_lintegral_meas_lt_mul lintegral_comp_eq_lintegral_meas_lt_mul
+#align lintegral_comp_eq_lintegral_meas_lt_mul MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul
 
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
-For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
+For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0..∞, μ {ω | f(ω) > t}`.
 
 See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ]
+theorem lintegral_eq_lintegral_meas_lt (μ : Measure α)
     (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) :
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t < f a} := by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+  filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
     with t ht
   rw [ht]
-#align lintegral_eq_lintegral_meas_lt lintegral_eq_lintegral_meas_lt
+#align lintegral_eq_lintegral_meas_lt MeasureTheory.lintegral_eq_lintegral_meas_lt
 
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
-For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
+For a nonnegative function `f` on a measure space, the Lebesgue integral of `f` can
 be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) > t}`.
 
 See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ]
+theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α)
     (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) := by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
   apply congr_arg fun z => ENNReal.ofReal p * z
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+  filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
     with t ht
   rw [ht]
-#align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
+#align lintegral_rpow_eq_lintegral_meas_lt_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul
 
 end LayercakeLT
 
 section LayercakeIntegral
 
-namespace MeasureTheory
-
 variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {f : α → ℝ}
 
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
@@ -400,34 +577,15 @@ See `lintegral_eq_lintegral_meas_lt` for a version with Lebesgue integral `∫
 theorem Integrable.integral_eq_integral_meas_lt
     (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
     (∫ ω, f ω ∂μ) = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by
-  obtain ⟨s, ⟨_, f_ae_zero_outside, s_sigmafin⟩⟩ :=
-    f_intble.aefinStronglyMeasurable.exists_set_sigmaFinite
-  have f_nn' : 0 ≤ᵐ[μ.restrict s] f := ae_restrict_of_ae f_nn
-  have f_intble' : Integrable f (μ.restrict s) := f_intble.restrict
-  have f_aemble' : AEMeasurable f (μ.restrict s) := f_intble.aemeasurable.restrict
-  rw [(set_integral_eq_integral_of_ae_restrict_eq_zero f_ae_zero_outside).symm]
-  have obs : ∀ t ∈ Ioi (0 : ℝ), (μ {a : α | t < f a}) = ((μ.restrict s) {a : α | t < f a}) := by
-    intro t ht
-    convert NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
-              f_intble.restrict.aemeasurable.nullMeasurable f_ae_zero_outside measurableSet_Ioi ?_
-    simp only [mem_Ioi, not_lt] at ht ⊢
-    exact ht.le
-  have obs' := @set_integral_congr ℝ ℝ _ _ (fun t ↦ ENNReal.toReal (μ {a : α | t < f a}))
-          (fun t ↦ ENNReal.toReal ((μ.restrict s) {a : α | t < f a})) _ (volume : Measure ℝ) _
-          (measurableSet_Ioi (a := (0 : ℝ)))
-          (fun x x_in_Ioi ↦ congrArg ENNReal.toReal (obs x x_in_Ioi))
-  rw [obs']
-  have key := lintegral_eq_lintegral_meas_lt (μ.restrict s) f_nn' f_aemble'
-  have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂(μ.restrict s) < ∞ :=
-    Integrable.lintegral_lt_top f_intble'
-  have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, (μ.restrict s) {a | t < f a} < ∞ :=
-    by simp only [← key, lhs_finite]
-  have rhs_integrand_finite : ∀ (t : ℝ), t > 0 → (μ.restrict s) {a | t < f a} < ∞ :=
-    fun t ht ↦ Integrable.measure_gt_lt_top f_intble' ht
+  have key := lintegral_eq_lintegral_meas_lt μ f_nn f_intble.aemeasurable
+  have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞ := Integrable.lintegral_lt_top f_intble
+  have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} < ∞ := by simp only [← key, lhs_finite]
+  have rhs_integrand_finite : ∀ (t : ℝ), t > 0 → μ {a | t < f a} < ∞ :=
+    fun t ht ↦ measure_gt_lt_top f_intble ht
   convert (ENNReal.toReal_eq_toReal lhs_finite.ne rhs_finite.ne).mpr key
-  · exact integral_eq_lintegral_of_nonneg_ae f_nn' f_intble'.aestronglyMeasurable
+  · exact integral_eq_lintegral_of_nonneg_ae f_nn f_intble.aestronglyMeasurable
   · have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure ℝ).restrict (Set.Ioi 0))
-      (fun t ↦ ENNReal.toReal ((μ.restrict s) {a : α | t < f a})) ?_ ?_
+      (fun t ↦ ENNReal.toReal (μ {a : α | t < f a})) ?_ ?_
     · rw [aux]
       congr 1
       apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall _)
@@ -437,6 +595,6 @@ theorem Integrable.integral_eq_integral_meas_lt
       refine Measurable.ennreal_toReal ?_
       exact Antitone.measurable (fun _ _ hst ↦ measure_mono (fun _ h ↦ lt_of_le_of_lt hst h))
 
-end MeasureTheory
-
 end LayercakeIntegral
+
+end MeasureTheory

Layer cake formulas currently exist for ENNReal-valued functions and Lebesgue integrals. This PR adds the most common version of the layer cake formula for integrable a.e.-nonnegative real-valued functions and Bochner integrals.

Co-authored-by: kkytola <“kalle.kytola@aalto.fi”> Co-authored-by: kkytola <39528102+kkytola@users.noreply.github.com>

@@ -5,6 +5,7 @@ Authors: Kalle Kytölä
 -/
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
 import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
 
 #align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
 
@@ -48,6 +49,12 @@ are also included.
    Other common special cases of the layer cake formulas, stating that for a nonnegative function f
    and p > 0, we have ∫ f(ω)^p ∂μ(ω) = p * ∫ μ {ω | f(ω) ≥ t} * t^(p-1) dt and
    ∫ f(ω)^p ∂μ(ω) = p * ∫ μ {ω | f(ω) > t} * t^(p-1) dt, respectively.
+ * `integral_eq_integral_meas_lt`:
+   A Bochner integral version of the most common special case of the layer cake formulas, stating
+   that for an integrable and a.e.-nonnegative function f we have
+   ∫ f(ω) ∂μ(ω) = ∫ μ {ω | f(ω) > t} dt. In this result, sigma-finiteness of μ does not need to be
+   explicitly assumed, because integrability guarantees sigma-finiteness of the restriction of μ
+   to the support of f.
 
 ## Tags
 
@@ -377,3 +384,59 @@ theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite 
 #align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
 
 end LayercakeLT
+
+section LayercakeIntegral
+
+namespace MeasureTheory
+
+variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {f : α → ℝ}
+
+/-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
+
+For an integrable a.e.-nonnegative real-valued function `f`, the Bochner integral of `f` can be
+written (roughly speaking) as: `∫ f ∂μ = ∫ t in 0..∞, μ {ω | f(ω) > t}`.
+
+See `lintegral_eq_lintegral_meas_lt` for a version with Lebesgue integral `∫⁻` instead. -/
+theorem Integrable.integral_eq_integral_meas_lt
+    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
+    (∫ ω, f ω ∂μ) = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by
+  obtain ⟨s, ⟨_, f_ae_zero_outside, s_sigmafin⟩⟩ :=
+    f_intble.aefinStronglyMeasurable.exists_set_sigmaFinite
+  have f_nn' : 0 ≤ᵐ[μ.restrict s] f := ae_restrict_of_ae f_nn
+  have f_intble' : Integrable f (μ.restrict s) := f_intble.restrict
+  have f_aemble' : AEMeasurable f (μ.restrict s) := f_intble.aemeasurable.restrict
+  rw [(set_integral_eq_integral_of_ae_restrict_eq_zero f_ae_zero_outside).symm]
+  have obs : ∀ t ∈ Ioi (0 : ℝ), (μ {a : α | t < f a}) = ((μ.restrict s) {a : α | t < f a}) := by
+    intro t ht
+    convert NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
+              f_intble.restrict.aemeasurable.nullMeasurable f_ae_zero_outside measurableSet_Ioi ?_
+    simp only [mem_Ioi, not_lt] at ht ⊢
+    exact ht.le
+  have obs' := @set_integral_congr ℝ ℝ _ _ (fun t ↦ ENNReal.toReal (μ {a : α | t < f a}))
+          (fun t ↦ ENNReal.toReal ((μ.restrict s) {a : α | t < f a})) _ (volume : Measure ℝ) _
+          (measurableSet_Ioi (a := (0 : ℝ)))
+          (fun x x_in_Ioi ↦ congrArg ENNReal.toReal (obs x x_in_Ioi))
+  rw [obs']
+  have key := lintegral_eq_lintegral_meas_lt (μ.restrict s) f_nn' f_aemble'
+  have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂(μ.restrict s) < ∞ :=
+    Integrable.lintegral_lt_top f_intble'
+  have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, (μ.restrict s) {a | t < f a} < ∞ :=
+    by simp only [← key, lhs_finite]
+  have rhs_integrand_finite : ∀ (t : ℝ), t > 0 → (μ.restrict s) {a | t < f a} < ∞ :=
+    fun t ht ↦ Integrable.measure_gt_lt_top f_intble' ht
+  convert (ENNReal.toReal_eq_toReal lhs_finite.ne rhs_finite.ne).mpr key
+  · exact integral_eq_lintegral_of_nonneg_ae f_nn' f_intble'.aestronglyMeasurable
+  · have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure ℝ).restrict (Set.Ioi 0))
+      (fun t ↦ ENNReal.toReal ((μ.restrict s) {a : α | t < f a})) ?_ ?_
+    · rw [aux]
+      congr 1
+      apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall _)
+      exact fun t t_pos ↦ ENNReal.ofReal_toReal (rhs_integrand_finite t t_pos).ne
+    · exact eventually_of_forall (fun x ↦ by simp only [Pi.zero_apply, ENNReal.toReal_nonneg])
+    · apply Measurable.aestronglyMeasurable
+      refine Measurable.ennreal_toReal ?_
+      exact Antitone.measurable (fun _ _ hst ↦ measure_mono (fun _ h ↦ lt_of_le_of_lt hst h))
+
+end MeasureTheory
+
+end LayercakeIntegral

The layercake formulas (a typical example of which is ∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) > t}) had been originally proven assuming measurability and nonnegativity of f. This PR generalizes them to null-measurable and a.e.-nonnegative f.

Co-authored-by: kkytola <“kalle.kytola@aalto.fi”> Co-authored-by: kkytola <39528102+kkytola@users.noreply.github.com>

@@ -78,8 +78,9 @@ See `MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul` and
 `lintegral_comp_eq_lintegral_meas_lt_mul` for the main formulations of the layer
 cake formula. -/
 theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α) [SigmaFinite μ]
-    (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
-    (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t) :
+    (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ)
+    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
+    (g_nn : ∀ t > 0, 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
   have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t := by
@@ -87,19 +88,19 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     cases' eq_or_lt_of_le ht with h h
     · simp [← h]
     · exact g_intble t h
-  have integrand_eq :
-    ∀ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) := by
-    intro ω
+  have integrand_eq : ∀ᵐ ω ∂μ,
+      ENNReal.ofReal (∫ t in (0)..f ω, g t) = (∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t)) := by
+    filter_upwards [f_nn] with ω fω_nn
     have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by
-      filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))] with x
-        hx using g_nn x hx.1
-    rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]
+      filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))]
+        with x hx using g_nn x hx.1
+    rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) fω_nn).1 g_ae_nn]
     congr
-    exact intervalIntegral.integral_of_le (f_nn ω)
-  simp_rw [integrand_eq, ← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]
+    exact intervalIntegral.integral_of_le fω_nn
+  rw [lintegral_congr_ae integrand_eq]
+  simp_rw [← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]
   -- Porting note: was part of `simp_rw` on the previous line, but didn't trigger.
-  rw [← lintegral_indicator _ measurableSet_Ioi]
-  rw [lintegral_lintegral_swap]
+  rw [← lintegral_indicator _ measurableSet_Ioi, lintegral_lintegral_swap]
   · apply congr_arg
     funext s
     have aux₁ :
@@ -113,8 +114,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
       · have h_copy := h
         simp only [mem_Ioc, not_and, not_le] at h
         by_cases h' : 0 < s
-        · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
-            mul_zero]
+        · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]
         · have : s ∉ Ioi (0 : ℝ) := h'
           simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,
             zero_mul, mem_Ioc, false_and_iff]
@@ -122,15 +122,13 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     rw [lintegral_const_mul']
     swap;
     · apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top
-      -- Porting note: was
-      -- by_cases s ∈ Ioi (0 : ℝ) <;> · simp [h]
-      by_cases h : (0 : ℝ) < s <;> · simp [indicator_apply, h]
+      by_cases h : (0 : ℝ) < s <;> · simp [h]
     simp_rw [show
         (fun a => (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
           {a : α | s ≤ f a}.indicator (fun _ => 1) a
         by funext a; by_cases s ≤ f a <;> simp [h]]
-    rw [lintegral_indicator]
-    swap; · exact f_mble measurableSet_Ici
+    rw [lintegral_indicator₀]
+    swap; · exact f_mble.nullMeasurable measurableSet_Ici
     rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,
       mul_assoc,
       show
@@ -152,9 +150,10 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
     · have h' : (p_fst, p_snd) ∉ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h
       rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]
   rw [aux₂]
-  have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ
-  simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble
-  exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator mble
+  have mble₀ : NullMeasurableSet {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} (μ.prod volume) := by
+    simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using
+      nullMeasurableSet_region_between_oc μ measurable_zero.aemeasurable f_mble MeasurableSet.univ
+  exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator₀ mble₀
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
 
 /-- The layer cake formula / Cavalieri's principle / tail probability formula:
@@ -165,12 +164,12 @@ with derivative `G' = g`. Then the integral of the composition `G ∘ f` can be
 the integral over the positive real line of the "tail measures" `μ {ω | f(ω) ≥ t}` of `f`
 weighted by `g`.
 
-Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in (0).. ∞, g(t) * μ {ω | f(ω) ≥ t}`.
+Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞, g(t) * μ {ω | f(ω) ≥ t}`.
 
 See `lintegral_comp_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
+theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+    (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
       ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
@@ -190,27 +189,29 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
     apply lintegral_congr_ae
     filter_upwards [g_eq_G] with a ha
     rw [ha]
-  have eq₂ : ∀ ω, (∫ t in (0)..f ω, g t) = ∫ t in (0)..f ω, G t := by
-    refine' fun ω => intervalIntegral.integral_congr_ae _
-    have fω_nn : 0 ≤ f ω := f_nn ω
+  have eq₂ : ∀ᵐ ω ∂μ,
+      ENNReal.ofReal (∫ t in (0)..f ω, g t) = ENNReal.ofReal (∫ t in (0)..f ω, G t) := by
+    filter_upwards [f_nn] with ω fω_nn
+    congr 1
+    refine' intervalIntegral.integral_congr_ae _
+    have fω_nn : 0 ≤ f ω := fω_nn
     rw [uIoc_of_le fω_nn, ←
       ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : ℝ) (f ω)))]
     exact g_eq_G_on (f ω)
-  simp_rw [eq₁, eq₂]
-  exact
-    lintegral_comp_eq_lintegral_meas_le_mul_of_measurable μ f_nn f_mble G_intble G_mble
-      fun t _ => G_nn t
+  simp_rw [lintegral_congr_ae eq₂, eq₁]
+  exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable μ f_nn f_mble
+          G_intble G_mble (fun t _ => G_nn t)
 #align measure_theory.lintegral_comp_eq_lintegral_meas_le_mul MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul
 
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
-be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in (0).. ∞, μ {ω | f(ω) ≥ t}`.
+be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0..∞, μ {ω | f(ω) ≥ t}`.
 
 See `lintegral_eq_lintegral_meas_lt` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) :
+theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+    (f_mble : AEMeasurable f μ) :
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
   set cst := fun _ : ℝ => (1 : ℝ)
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
@@ -227,12 +228,12 @@ theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
-be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in (0).. ∞, t^(p-1) * μ {ω | f(ω) ≥ t}`.
+be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) ≥ t}`.
 
 See `lintegral_rpow_eq_lintegral_meas_lt_mul` for a version with sets of the form `{ω | f(ω) > t}`
 instead. -/
-theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
+theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+    (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by
   have one_lt_p : -1 < p - 1 := by linarith
@@ -248,10 +249,13 @@ theorem lintegral_rpow_eq_lintegral_meas_le_mul (μ : Measure α) [SigmaFinite 
   have g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>
     intervalIntegral.intervalIntegrable_rpow' one_lt_p
   have key := lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn
-  rw [← key, ← lintegral_const_mul (ENNReal.ofReal p)] <;> simp_rw [obs]
+  rw [← key, ← lintegral_const_mul'' (ENNReal.ofReal p)] <;> simp_rw [obs]
   · congr with ω
     rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f ω ^ p) p_pos.ne.symm]
-  · exact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal
+  · have aux := (@measurable_const ℝ α (by infer_instance) (by infer_instance) p).aemeasurable
+                  (μ := μ)
+    exact (Measurable.ennreal_ofReal (hf := measurable_id)).comp_aemeasurable
+      ((f_mble.pow aux).div_const p)
 #align measure_theory.lintegral_rpow_eq_lintegral_meas_le_mul MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
 
 end MeasureTheory
@@ -268,35 +272,40 @@ variable {β : Type*} [MeasurableSpace β] [MeasurableSingletonClass β]
 
 namespace Measure
 
-theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type*} [LinearOrder R] [MeasurableSpace R]
-    [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
-    (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} := by
+theorem meas_eq_pos_of_meas_le_ne_meas_lt
+    {α : Type*} [MeasurableSpace α] {μ : Measure α} {R : Type*} [LinearOrder R]
+    {g : α → R} {t : R} (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) :
+    0 < μ {a : α | g a = t} := by
+  by_contra con
+  rw [not_lt, nonpos_iff_eq_zero] at con
+  apply ht
+  refine le_antisymm ?_ (measure_mono (fun a ha ↦ le_of_lt ha))
   have uni : {a : α | t ≤ g a} = {a : α | t < g a} ∪ {a : α | t = g a} := by
     ext a
-    simp only [mem_setOf, mem_union]
-    apply le_iff_lt_or_eq
+    simpa only [mem_setOf, mem_union] using le_iff_lt_or_eq
   rw [show {a : α | t = g a} = {a : α | g a = t} by simp_rw [eq_comm]] at uni
-  have disj : {a : α | t < g a} ∩ {a : α | g a = t} = ∅ := by
-    ext a
-    simp only [mem_inter_iff, mem_setOf, mem_empty_iff_false, iff_false_iff, not_and]
-    exact ne_of_gt
-  have μ_add : μ {a : α | t ≤ g a} = μ {a : α | t < g a} + μ {a : α | g a = t} := by
-    rw [uni,
-      measure_union (disjoint_iff_inter_eq_empty.mpr disj)
-        (g_mble (Finite.measurableSet (finite_singleton t)))]
-  by_contra con
-  rw [not_lt, nonpos_iff_eq_zero] at con
-  rw [con, add_zero] at μ_add
-  exact ht μ_add
-#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
+  have μ_le_add := measure_union_le (μ := μ) {a : α | t < g a} {a : α | g a = t}
+  rwa [con, add_zero, ← uni] at μ_le_add
+#align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_eq_pos_of_meas_le_ne_meas_lt
+
+theorem countable_meas_le_ne_meas_lt₀ [SigmaFinite μ] {R : Type*} [LinearOrder R]
+    [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : NullMeasurable g μ) :
+    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
+  Countable.mono (fun _ h ↦ meas_eq_pos_of_meas_le_ne_meas_lt h)
+    (Measure.countable_meas_level_set_pos₀ g_mble)
 
 theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
     {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
-  Countable.mono (show _ from fun _ ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
-    (Measure.countable_meas_level_set_pos g_mble)
+  countable_meas_le_ne_meas_lt₀ (μ := μ) g_mble.nullMeasurable
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
 
+theorem meas_le_ae_eq_meas_lt₀ [SigmaFinite μ] {R : Type*} [LinearOrder R] [MeasurableSpace R]
+    [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R}
+    (g_mble : NullMeasurable g μ) :
+    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
+  Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt₀ μ g_mble) _
+
 theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
     (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
@@ -315,52 +324,55 @@ with derivative `G' = g`. Then the integral of the composition `G ∘ f` can be
 the integral over the positive real line of the "tail measures" `μ {ω | f(ω) > t}` of `f`
 weighted by `g`.
 
-Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in (0).. ∞, g(t) * μ {ω | f(ω) > t}`.
+Roughly speaking, the statement is: `∫⁻ (G ∘ f) ∂μ = ∫⁻ t in 0..∞, g(t) * μ {ω | f(ω) > t}`.
 
 See `lintegral_comp_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
+theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ᵐ[μ] f)
+    (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
     (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
     (∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ) =
       ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) := by
   rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+    with t ht
   rw [ht]
 #align lintegral_comp_eq_lintegral_meas_lt_mul lintegral_comp_eq_lintegral_meas_lt_mul
 
 /-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
-be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in (0).. ∞, μ {ω | f(ω) > t}`.
+be written (roughly speaking) as: `∫⁻ f ∂μ = ∫⁻ t in 0..∞, μ {ω | f(ω) > t}`.
 
 See `lintegral_eq_lintegral_meas_le` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) :
+theorem lintegral_eq_lintegral_meas_lt (μ : Measure α) [SigmaFinite μ]
+    (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) :
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t < f a} := by
   rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble]
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+    with t ht
   rw [ht]
 #align lintegral_eq_lintegral_meas_lt lintegral_eq_lintegral_meas_lt
 
 /-- An application of the layer cake formula / Cavalieri's principle / tail probability formula:
 
 For a nonnegative function `f` on a sigma-finite measure space, the Lebesgue integral of `f` can
-be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in (0).. ∞, t^(p-1) * μ {ω | f(ω) > t}`.
+be written (roughly speaking) as: `∫⁻ f^p ∂μ = p * ∫⁻ t in 0..∞, t^(p-1) * μ {ω | f(ω) > t}`.
 
 See `lintegral_rpow_eq_lintegral_meas_le_mul` for a version with sets of the form `{ω | f(ω) ≥ t}`
 instead. -/
-theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
-    (f_mble : Measurable f) {p : ℝ} (p_pos : 0 < p) :
+theorem lintegral_rpow_eq_lintegral_meas_lt_mul (μ : Measure α) [SigmaFinite μ]
+    (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p) :
     (∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ) =
       ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) := by
   rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
   apply congr_arg fun z => ENNReal.ofReal p * z
   apply lintegral_congr_ae
-  filter_upwards [Measure.meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f_mble] with t ht
+  filter_upwards [Measure.meas_le_ae_eq_meas_lt₀ μ (volume.restrict (Ioi 0)) f_mble.nullMeasurable]
+    with t ht
   rw [ht]
 #align lintegral_rpow_eq_lintegral_meas_lt_mul lintegral_rpow_eq_lintegral_meas_lt_mul
 

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).

@@ -114,10 +114,10 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
         simp only [mem_Ioc, not_and, not_le] at h
         by_cases h' : 0 < s
         · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
-            MulZeroClass.mul_zero]
+            mul_zero]
         · have : s ∉ Ioi (0 : ℝ) := h'
-          simp only [this, h', indicator_of_not_mem, not_false_iff, MulZeroClass.mul_zero,
-            MulZeroClass.zero_mul, mem_Ioc, false_and_iff]
+          simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,
+            zero_mul, mem_Ioc, false_and_iff]
     simp_rw [aux₁]
     rw [lintegral_const_mul']
     swap;

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

This has nice performance benefits.

@@ -68,7 +68,7 @@ section Layercake
 
 namespace MeasureTheory
 
-variable {α : Type _} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
+variable {α : Type*} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} {s : Set α}
 
 /-- An auxiliary version of the layer cake formula (Cavalieri's principle, tail probability
 formula), with a measurability assumption that would also essentially follow from the
@@ -262,13 +262,13 @@ section LayercakeLT
 
 open MeasureTheory
 
-variable {α : Type _} [MeasurableSpace α] (μ : Measure α)
+variable {α : Type*} [MeasurableSpace α] (μ : Measure α)
 
-variable {β : Type _} [MeasurableSpace β] [MeasurableSingletonClass β]
+variable {β : Type*} [MeasurableSpace β] [MeasurableSingletonClass β]
 
 namespace Measure
 
-theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [MeasurableSpace R]
+theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type*} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) {t : R}
     (ht : μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}) : 0 < μ {a : α | g a = t} := by
   have uni : {a : α | t ≤ g a} = {a : α | t < g a} ∪ {a : α | t = g a} := by
@@ -290,14 +290,14 @@ theorem meas_le_ne_meas_lt_subset_meas_pos {R : Type _} [LinearOrder R] [Measura
   exact ht μ_add
 #align measure.meas_le_ne_meas_lt_subset_meas_pos Measure.meas_le_ne_meas_lt_subset_meas_pos
 
-theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R]
+theorem countable_meas_le_ne_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R]
     [MeasurableSpace R] [MeasurableSingletonClass R] {g : α → R} (g_mble : Measurable g) :
     {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable :=
   Countable.mono (show _ from fun _ ht => meas_le_ne_meas_lt_subset_meas_pos μ g_mble ht)
     (Measure.countable_meas_level_set_pos g_mble)
 #align measure.countable_meas_le_ne_meas_lt Measure.countable_meas_le_ne_meas_lt
 
-theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type _} [LinearOrder R] [MeasurableSpace R]
+theorem meas_le_ae_eq_meas_lt [SigmaFinite μ] {R : Type*} [LinearOrder R] [MeasurableSpace R]
     [MeasurableSingletonClass R] (ν : Measure R) [NoAtoms ν] {g : α → R} (g_mble : Measurable g) :
     (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} :=
   Set.Countable.measure_zero (Measure.countable_meas_le_ne_meas_lt μ g_mble) _

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Kalle Kytölä. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
-
-! This file was ported from Lean 3 source module measure_theory.integral.layercake
-! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
 import Mathlib.Analysis.SpecialFunctions.Integrals
 
+#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
+
 /-!
 # The layer cake formula / Cavalieri's principle / tail probability formula
 

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.

@@ -111,14 +111,12 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
           (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f x) := by
       funext a
       by_cases s ∈ Ioc (0 : ℝ) (f a)
-      ·
-        simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
+      · simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
           mul_one]
       · have h_copy := h
         simp only [mem_Ioc, not_and, not_le] at h
         by_cases h' : 0 < s
-        ·
-          simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
+        · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,
             MulZeroClass.mul_zero]
         · have : s ∉ Ioi (0 : ℝ) := h'
           simp only [this, h', indicator_of_not_mem, not_false_iff, MulZeroClass.mul_zero,

Fix the set tactic to not time out when dealing with slow to elaborate terms and many local hypotheses.

The root cause of this is that the rewrite [blah] at * tactic causes blah to be elaborated again and again for each local hypothesis, this is possibly a core issue that should be fixed separately, but in set we have the elaborated term already so we can just use it.

We also add some functionality to simply test / demonstrate failures when elaboration takes too long, namely sleepAtLeastHeartbeats and a sleep_heartbeats tactic.

@urkud was facing some slow running set's in https://github.com/leanprover-community/mathlib4/pull/4912, see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Timeout.20in.20.60set.20.2E.2E.20with.60/near/367958828 that this issue was minimized from and should fix.

Some other linter failures show up due to changes to the set internals so fix these too.

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

@@ -217,7 +217,7 @@ instead. -/
 theorem lintegral_eq_lintegral_meas_le (μ : Measure α) [SigmaFinite μ] (f_nn : 0 ≤ f)
     (f_mble : Measurable f) :
     (∫⁻ ω, ENNReal.ofReal (f ω) ∂μ) = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} := by
-  set cst := fun t : ℝ => (1 : ℝ)
+  set cst := fun _ : ℝ => (1 : ℝ)
   have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
     intervalIntegrable_const
   have key :=