measure_theory.integral.integral_eq_improper ⟷ Mathlib.MeasureTheory.Integral.IntegralEqImproper

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

@@ -722,8 +722,8 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
 -/
 
-#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded /-
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
+#print MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded /-
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
     (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, ∫ x in Ioc (a i) (b i), ‖f x‖ ≤ I) :
     IntegrableOn f (Ioc a₀ b₀) :=
@@ -735,23 +735,23 @@ theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
   refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi
   · apply ae_of_all; simp only [Pi.zero_apply, norm_nonneg, forall_const]
   · apply ae_of_all; intro c hc; exact hc.1
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
 -/
 
-#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left /-
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
+#print MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left /-
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
     (h : ∀ᶠ i in l, ∫ x in Ioc (a i) b, ‖f x‖ ≤ I) : IntegrableOn f (Ioc a₀ b) :=
-  integrableOn_Ioc_of_interval_integral_norm_bounded hfi ha tendsto_const_nhds h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi ha tendsto_const_nhds h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left
 -/
 
-#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right /-
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded_right {I a b₀ : ℝ}
+#print MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right /-
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
     (h : ∀ᶠ i in l, ∫ x in Ioc a (b i), ‖f x‖ ≤ I) : IntegrableOn f (Ioc a b₀) :=
-  integrableOn_Ioc_of_interval_integral_norm_bounded hfi tendsto_const_nhds hb h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi tendsto_const_nhds hb h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right
 -/
 
 end IntegrableOfIntervalIntegral

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

@@ -1079,7 +1079,7 @@ theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 <
   rw [← integral_indicator (this a), ← integral_indicator (this <| b * a)]
   convert measure.integral_comp_mul_left _ b
   ext1 x
-  rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left _ hb.ne']
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left₀ _ hb.ne']
 #align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
 -/
 
@@ -1148,7 +1148,8 @@ theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (
   rw [← integrable_indicator_iff (measurableSet_Ioi : MeasurableSet <| Ioi <| a * c)]
   convert integrable_comp_mul_left_iff ((Ioi (a * c)).indicator f) ha.ne' using 2
   ext1 x
-  rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c, mul_div_cancel _ ha.ne']
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c,
+    mul_div_cancel_right₀ _ ha.ne']
 #align measure_theory.integrable_on_Ioi_comp_mul_left_iff MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
 -/
 

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

@@ -363,7 +363,7 @@ theorem AECover.aemeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
-  rwa [measure.restrict_eq_self_of_ae_mem] at this 
+  rwa [measure.restrict_eq_self_of_ae_mem] at this
   filter_upwards [hφ.ae_eventually_mem] with x hx using
     let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
@@ -377,7 +377,7 @@ theorem AECover.aestronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_strongly_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
-  rwa [measure.restrict_eq_self_of_ae_mem] at this 
+  rwa [measure.restrict_eq_self_of_ae_mem] at this
   filter_upwards [hφ.ae_eventually_mem] with x hx using
     let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
@@ -1007,7 +1007,7 @@ theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : 
         (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2)
     · rw [min_eq_left hb.le]; exact Ioo_subset_Ioi_self
     · rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self
-  rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2 
+  rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2
   have t2 := interval_integral_tendsto_integral_Ioi _ hg2 tendsto_id
   have : Ioi (f a) ⊆ f '' Ici a :=
     Ioi_subset_Ici_self.trans <|
@@ -1055,7 +1055,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g
-  rw [a3] at this ; rw [this]
+  rw [a3] at this; rw [this]
   refine' set_integral_congr measurableSet_Ioi _
   intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
@@ -1123,7 +1123,7 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
     · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integrable_on_image_iff_integrable_on_abs_deriv_smul measurableSet_Ioi a1 a2 f
-  rw [a3] at this 
+  rw [a3] at this
   rw [this]
   refine' integrable_on_congr_fun (fun x hx => _) measurableSet_Ioi
   simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]

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

@@ -830,7 +830,7 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
   apply
     intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self)
       fun y hy => hderiv y hy.1
-  rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x]
   exact f'int.mono (fun y hy => hy.1) le_rfl
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
 -/
@@ -872,7 +872,7 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
       apply
         intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self)
           fun y hy => hderiv y hy.1
-      rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+      rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x]
       exact
         intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
           (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1

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

@@ -3,11 +3,11 @@ Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 -/
-import Mathbin.Analysis.SpecialFunctions.Pow.Deriv
-import Mathbin.MeasureTheory.Integral.FundThmCalculus
-import Mathbin.Order.Filter.AtTopBot
-import Mathbin.MeasureTheory.Function.Jacobian
-import Mathbin.MeasureTheory.Measure.Haar.NormedSpace
+import Analysis.SpecialFunctions.Pow.Deriv
+import MeasureTheory.Integral.FundThmCalculus
+import Order.Filter.AtTopBot
+import MeasureTheory.Function.Jacobian
+import MeasureTheory.Measure.Haar.NormedSpace
 
 #align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
-
-! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! 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.Analysis.SpecialFunctions.Pow.Deriv
 import Mathbin.MeasureTheory.Integral.FundThmCalculus
@@ -14,6 +9,8 @@ import Mathbin.Order.Filter.AtTopBot
 import Mathbin.MeasureTheory.Function.Jacobian
 import Mathbin.MeasureTheory.Measure.Haar.NormedSpace
 
+#align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
 /-!
 # Links between an integral and its "improper" version
 

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

@@ -106,6 +106,7 @@ section Preorderα
 variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
+#print MeasureTheory.aecover_Icc /-
 theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
@@ -113,18 +114,23 @@ theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) :=
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Icc }
 #align measure_theory.ae_cover_Icc MeasureTheory.aecover_Icc
+-/
 
+#print MeasureTheory.aecover_Ici /-
 theorem aecover_Ici : AECover μ l fun i => Ici <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_le_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ici }
 #align measure_theory.ae_cover_Ici MeasureTheory.aecover_Ici
+-/
 
+#print MeasureTheory.aecover_Iic /-
 theorem aecover_Iic : AECover μ l fun i => Iic <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iic }
 #align measure_theory.ae_cover_Iic MeasureTheory.aecover_Iic
+-/
 
 end Preorderα
 
@@ -133,6 +139,7 @@ section LinearOrderα
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
+#print MeasureTheory.aecover_Ioo /-
 theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
@@ -140,7 +147,9 @@ theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioo }
 #align measure_theory.ae_cover_Ioo MeasureTheory.aecover_Ioo
+-/
 
+#print MeasureTheory.aecover_Ioc /-
 theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
@@ -148,7 +157,9 @@ theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) :=
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioc }
 #align measure_theory.ae_cover_Ioc MeasureTheory.aecover_Ioc
+-/
 
+#print MeasureTheory.aecover_Ico /-
 theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
@@ -156,18 +167,23 @@ theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) :=
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ico }
 #align measure_theory.ae_cover_Ico MeasureTheory.aecover_Ico
+-/
 
+#print MeasureTheory.aecover_Ioi /-
 theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_lt_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ioi }
 #align measure_theory.ae_cover_Ioi MeasureTheory.aecover_Ioi
+-/
 
+#print MeasureTheory.aecover_Iio /-
 theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iio }
 #align measure_theory.ae_cover_Iio MeasureTheory.aecover_Iio
+-/
 
 end LinearOrderα
 
@@ -176,6 +192,7 @@ section FiniteIntervals
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B))
 
+#print MeasureTheory.aecover_Ioo_of_Icc /-
 theorem aecover_Ioo_of_Icc : AECover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
@@ -184,7 +201,9 @@ theorem aecover_Ioo_of_Icc : AECover (μ.restrict <| Ioo A B) l fun i => Icc (a
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Icc }
 #align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aecover_Ioo_of_Icc
+-/
 
+#print MeasureTheory.aecover_Ioo_of_Ico /-
 theorem aecover_Ioo_of_Ico : AECover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
@@ -193,7 +212,9 @@ theorem aecover_Ioo_of_Ico : AECover (μ.restrict <| Ioo A B) l fun i => Ico (a
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ico }
 #align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aecover_Ioo_of_Ico
+-/
 
+#print MeasureTheory.aecover_Ioo_of_Ioc /-
 theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
@@ -202,7 +223,9 @@ theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <| Ioo A B) l fun i => Ioc (a
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioc }
 #align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aecover_Ioo_of_Ioc
+-/
 
+#print MeasureTheory.aecover_Ioo_of_Ioo /-
 theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
@@ -211,6 +234,7 @@ theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <| Ioo A B) l fun i => Ioo (a
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioo }
 #align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aecover_Ioo_of_Ioo
+-/
 
 variable [NoAtoms μ]
 
@@ -300,32 +324,40 @@ theorem aecover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B
 
 end FiniteIntervals
 
+#print MeasureTheory.AECover.restrict /-
 theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} :
     AECover (μ.restrict s) l φ :=
   { ae_eventually_mem := ae_restrict_of_ae hφ.ae_eventually_mem
     Measurable := hφ.Measurable }
 #align measure_theory.ae_cover.restrict MeasureTheory.AECover.restrict
+-/
 
+#print MeasureTheory.aecover_restrict_of_ae_imp /-
 theorem aecover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
     (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n)
     (measurable : ∀ n, MeasurableSet <| φ n) : AECover (μ.restrict s) l φ :=
   { ae_eventually_mem := by rwa [ae_restrict_iff' hs]
     Measurable }
 #align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aecover_restrict_of_ae_imp
+-/
 
+#print MeasureTheory.AECover.inter_restrict /-
 theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α}
     (hs : MeasurableSet s) : AECover (μ.restrict s) l fun i => φ i ∩ s :=
   aecover_restrict_of_ae_imp hs
     (hφ.ae_eventually_mem.mono fun x hx hxs => hx.mono fun i hi => ⟨hi, hxs⟩) fun i =>
     (hφ.Measurable i).inter hs
 #align measure_theory.ae_cover.inter_restrict MeasureTheory.AECover.inter_restrict
+-/
 
+#print MeasureTheory.AECover.ae_tendsto_indicator /-
 theorem AECover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace β] (f : α → β)
     {φ : ι → Set α} (hφ : AECover μ l φ) :
     ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <| f x) :=
   hφ.ae_eventually_mem.mono fun x hx =>
     tendsto_const_nhds.congr' <| hx.mono fun n hn => (indicator_of_mem hn _).symm
 #align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AECover.ae_tendsto_indicator
+-/
 
 #print MeasureTheory.AECover.aemeasurable /-
 theorem AECover.aemeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.ne_bot]
@@ -357,12 +389,14 @@ theorem AECover.aestronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
 
 end AeCover
 
+#print MeasureTheory.AECover.comp_tendsto /-
 theorem AECover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
     {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
     AECover μ l' (φ ∘ u) :=
   { ae_eventually_mem := hφ.ae_eventually_mem.mono fun x hx => hu.Eventually hx
     Measurable := fun i => hφ.Measurable (u i) }
 #align measure_theory.ae_cover.comp_tendsto MeasureTheory.AECover.comp_tendsto
+-/
 
 section AeCoverUnionInterCountable
 
@@ -377,6 +411,7 @@ theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AE
 #align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AECover.biUnion_Iic_aecover
 -/
 
+#print MeasureTheory.AECover.biInter_Ici_aecover /-
 theorem AECover.biInter_Ici_aecover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
     (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (h : k ∈ Ici n), φ k :=
   { ae_eventually_mem :=
@@ -390,6 +425,7 @@ theorem AECover.biInter_Ici_aecover [SemilatticeSup ι] [Nonempty ι] {φ : ι 
           exact mem_bInter fun k hk => hi k (le_trans hj hk))
     Measurable := fun i => MeasurableSet.biInter (to_countable _) fun n _ => hφ.Measurable n }
 #align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AECover.biInter_Ici_aecover
+-/
 
 end AeCoverUnionInterCountable
 
@@ -409,6 +445,7 @@ private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ
   (lintegral_tendsto_of_tendsto_of_monotone key₁ key₂ key₃).congr fun n =>
     lintegral_indicator f (hφ.Measurable n)
 
+#print MeasureTheory.AECover.lintegral_tendsto_of_nat /-
 theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
   by
@@ -422,19 +459,25 @@ theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ
   have le₂ := fun n => lintegral_mono_set (subset_bUnion_of_mem right_mem_Iic)
   exact tendsto_of_tendsto_of_tendsto_of_le_of_le lim₁ lim₂ le₁ le₂
 #align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AECover.lintegral_tendsto_of_nat
+-/
 
+#print MeasureTheory.AECover.lintegral_tendsto_of_countably_generated /-
 theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
     Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <| ∫⁻ x, f x ∂μ) :=
   tendsto_of_seq_tendsto fun u hu => (hφ.comp_tendsto hu).lintegral_tendsto_of_nat hfm
 #align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AECover.lintegral_tendsto_of_countably_generated
+-/
 
+#print MeasureTheory.AECover.lintegral_eq_of_tendsto /-
 theorem AECover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
 #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto
+-/
 
+#print MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated /-
 theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
     [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ :=
@@ -447,6 +490,7 @@ theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot
   rcases(eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
   exact ⟨i, lt_of_lt_of_le hm₁ hi⟩
 #align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated
+-/
 
 end Lintegral
 
@@ -454,6 +498,7 @@ section Integrable
 
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
+#print MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded /-
 theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖₊ ∂μ ≤ ENNReal.ofReal I) : Integrable f μ :=
@@ -461,7 +506,9 @@ theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountably
   refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
   exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded
+-/
 
+#print MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto /-
 theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
@@ -472,21 +519,27 @@ theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountably
   refine' (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _
   exact lt_max_of_lt_right (lt_add_one I)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto
+-/
 
+#print MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded' /-
 theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖₊ ∂μ ≤ I) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
+-/
 
+#print MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto' /-
 theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_tendsto I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto'
+-/
 
+#print MeasureTheory.AECover.integrable_of_integral_norm_bounded /-
 theorem AECover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hbounded : ∀ᶠ i in l, ∫ x in φ i, ‖f x‖ ∂μ ≤ I) : Integrable f μ :=
@@ -505,14 +558,18 @@ theorem AECover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGen
   rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)]
   apply ENNReal.ofReal_le_ofReal hi
 #align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AECover.integrable_of_integral_norm_bounded
+-/
 
+#print MeasureTheory.AECover.integrable_of_integral_norm_tendsto /-
 theorem AECover.integrable_of_integral_norm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ :=
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
   hφ.integrable_of_integral_norm_bounded I' hfi hI'
 #align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AECover.integrable_of_integral_norm_tendsto
+-/
 
+#print MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae /-
 theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, ∫ x in φ i, f x ∂μ ≤ I) : Integrable f μ :=
@@ -520,7 +577,9 @@ theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCoun
     hbounded.mono fun i hi =>
       (integral_congr_ae <| ae_restrict_of_ae <| hnng.mono fun x => Real.norm_of_nonneg).le.trans hi
 #align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
+-/
 
+#print MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae /-
 theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) :
@@ -528,6 +587,7 @@ theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCoun
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
   hφ.integrable_of_integral_bounded_of_nonneg_ae I' hfi hnng hI'
 #align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae
+-/
 
 end Integrable
 
@@ -536,6 +596,7 @@ section Integral
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
   [NormedSpace ℝ E] [CompleteSpace E]
 
+#print MeasureTheory.AECover.integral_tendsto_of_countably_generated /-
 theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) :
     Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) :=
@@ -546,7 +607,9 @@ theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated]
     (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm
     (hφ.ae_tendsto_indicator f)
 #align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AECover.integral_tendsto_of_countably_generated
+-/
 
+#print MeasureTheory.AECover.integral_eq_of_tendsto /-
 /-- Slight reformulation of
     `measure_theory.ae_cover.integral_tendsto_of_countably_generated`. -/
 theorem AECover.integral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
@@ -554,7 +617,9 @@ theorem AECover.integral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ :
     (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h
 #align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto
+-/
 
+#print MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae /-
 theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
     (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
@@ -562,6 +627,7 @@ theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGen
   have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto
   hφ.integral_eq_of_tendsto I hfi' htendsto
 #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
+-/
 
 end Integral
 
@@ -570,6 +636,7 @@ section IntegrableOfIntervalIntegral
 variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l]
   [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E}
 
+#print MeasureTheory.integrable_of_intervalIntegral_norm_bounded /-
 theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
     (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, ∫ x in a i..b i, ‖f x‖ ∂μ ≤ I) : Integrable f μ :=
@@ -580,6 +647,7 @@ theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
     hb.eventually (eventually_ge_at_top 0)] with i hai hbi ht
   rwa [← intervalIntegral.integral_of_le (hai.trans hbi)]
 #align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded
+-/
 
 #print MeasureTheory.integrable_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc (a i) (b i)`,
@@ -595,6 +663,7 @@ theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
 #align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrable_of_intervalIntegral_norm_tendsto
 -/
 
+#print MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded /-
 theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
     (h : ∀ᶠ i in l, ∫ x in a i..b, ‖f x‖ ∂μ ≤ I) : IntegrableOn f (Iic b) μ :=
@@ -610,6 +679,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
   rw [intervalIntegral.integral_of_le hai, measure.restrict_restrict (hφ.measurable i)]
   exact id
 #align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
+-/
 
 #print MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc (a i) b`,
@@ -624,6 +694,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
 #align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto
 -/
 
+#print MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded /-
 theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
     (h : ∀ᶠ i in l, ∫ x in a..b i, ‖f x‖ ∂μ ≤ I) : IntegrableOn f (Ioi a) μ :=
@@ -639,6 +710,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
   rw [intervalIntegral.integral_of_le hbi, measure.restrict_restrict (hφ.measurable i), inter_comm]
   exact id
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
+-/
 
 #print MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc a (b i)`,
@@ -653,6 +725,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
 -/
 
+#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded /-
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
     (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, ∫ x in Ioc (a i) (b i), ‖f x‖ ≤ I) :
@@ -666,18 +739,23 @@ theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
   · apply ae_of_all; simp only [Pi.zero_apply, norm_nonneg, forall_const]
   · apply ae_of_all; intro c hc; exact hc.1
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded
+-/
 
+#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left /-
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
     (h : ∀ᶠ i in l, ∫ x in Ioc (a i) b, ‖f x‖ ≤ I) : IntegrableOn f (Ioc a₀ b) :=
   integrableOn_Ioc_of_interval_integral_norm_bounded hfi ha tendsto_const_nhds h
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left
+-/
 
+#print MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right /-
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded_right {I a b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
     (h : ∀ᶠ i in l, ∫ x in Ioc a (b i), ‖f x‖ ≤ I) : IntegrableOn f (Ioc a b₀) :=
   integrableOn_Ioc_of_interval_integral_norm_bounded hfi tendsto_const_nhds hb h
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
+-/
 
 end IntegrableOfIntervalIntegral
 
@@ -738,6 +816,7 @@ section IoiFTC
 variable {E : Type _} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
   [NormedSpace ℝ E] [CompleteSpace E]
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto /-
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
 When a function has a limit at infinity `m`, and its derivative is integrable, then the
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
@@ -757,7 +836,9 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
   rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
   exact f'int.mono (fun y hy => hy.1) le_rfl
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
+-/
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto' /-
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
 When a function has a limit at infinity `m`, and its derivative is integrable, then the
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
@@ -769,7 +850,9 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDer
   intro x hx
   exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto' MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto'
+-/
 
+#print MeasureTheory.integrableOn_Ioi_deriv_of_nonneg /-
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `(a, +∞)` and continuity on `[a, +∞)`. -/
@@ -804,7 +887,9 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
       rw [abs_of_nonneg]
       exact g'pos _ hy.1
 #align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
+-/
 
+#print MeasureTheory.integrableOn_Ioi_deriv_of_nonneg' /-
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
@@ -815,7 +900,9 @@ theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt
   intro x hx
   exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
 #align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOn_Ioi_deriv_of_nonneg'
+-/
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg /-
 /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
@@ -826,7 +913,9 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
+-/
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg' /-
 /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
@@ -835,7 +924,9 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDeri
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonneg' hderiv g'pos hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
+-/
 
+#print MeasureTheory.integrableOn_Ioi_deriv_of_nonpos /-
 /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `(a, +∞)` and continuity on `[a, +∞)`. -/
@@ -848,7 +939,9 @@ theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousOn g (Ici a))
     integrable_on_Ioi_deriv_of_nonneg hcont.neg (fun x hx => (hderiv x hx).neg)
       (fun x hx => neg_nonneg_of_nonpos (g'neg x hx)) hg.neg
 #align measure_theory.integrable_on_Ioi_deriv_of_nonpos MeasureTheory.integrableOn_Ioi_deriv_of_nonpos
+-/
 
+#print MeasureTheory.integrableOn_Ioi_deriv_of_nonpos' /-
 /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
@@ -859,7 +952,9 @@ theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt
   intro x hx
   exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
 #align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
+-/
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos /-
 /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
@@ -870,7 +965,9 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonpos hcont hderiv g'neg hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
+-/
 
+#print MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos' /-
 /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
@@ -879,6 +976,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDeri
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
+-/
 
 end IoiFTC
 
@@ -937,6 +1035,7 @@ theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a :
 #align measure_theory.integral_comp_mul_deriv_Ioi MeasureTheory.integral_comp_mul_deriv_Ioi
 -/
 
+#print MeasureTheory.integral_comp_rpow_Ioi /-
 /-- Substitution `y = x ^ p` in integrals over `Ioi 0` -/
 theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     ∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ y in Ioi 0, g y :=
@@ -964,14 +1063,18 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
   intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
 #align measure_theory.integral_comp_rpow_Ioi MeasureTheory.integral_comp_rpow_Ioi
+-/
 
+#print MeasureTheory.integral_comp_rpow_Ioi_of_pos /-
 theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
     ∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ y in Ioi 0, g y :=
   by
   convert integral_comp_rpow_Ioi g hp.ne'
   funext; congr; rw [abs_of_nonneg hp.le]
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
+-/
 
+#print MeasureTheory.integral_comp_mul_left_Ioi /-
 theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
     ∫ x in Ioi a, g (b * x) = |b⁻¹| • ∫ x in Ioi (b * a), g x :=
   by
@@ -981,11 +1084,14 @@ theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 <
   ext1 x
   rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left _ hb.ne']
 #align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
+-/
 
+#print MeasureTheory.integral_comp_mul_right_Ioi /-
 theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
     ∫ x in Ioi a, g (x * b) = |b⁻¹| • ∫ x in Ioi (a * b), g x := by
   simpa only [mul_comm] using integral_comp_mul_left_Ioi g a hb
 #align measure_theory.integral_comp_mul_right_Ioi MeasureTheory.integral_comp_mul_right_Ioi
+-/
 
 end IoiChangeVariables
 
@@ -997,6 +1103,7 @@ open scoped Interval
 
 variable {E : Type _} [NormedAddCommGroup E]
 
+#print MeasureTheory.integrableOn_Ioi_comp_rpow_iff /-
 /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability. -/
 theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) :=
@@ -1024,7 +1131,9 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
   refine' integrable_on_congr_fun (fun x hx => _) measurableSet_Ioi
   simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
 #align measure_theory.integrable_on_Ioi_comp_rpow_iff MeasureTheory.integrableOn_Ioi_comp_rpow_iff
+-/
 
+#print MeasureTheory.integrableOn_Ioi_comp_rpow_iff' /-
 /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability (version
 without `|p|` factor) -/
 theorem integrableOn_Ioi_comp_rpow_iff' [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
@@ -1032,7 +1141,9 @@ theorem integrableOn_Ioi_comp_rpow_iff' [NormedSpace ℝ E] (f : ℝ → E) {p :
   simpa only [← integrable_on_Ioi_comp_rpow_iff f hp, mul_smul] using
     (integrable_smul_iff (abs_pos.mpr hp).ne' _).symm
 #align measure_theory.integrable_on_Ioi_comp_rpow_iff' MeasureTheory.integrableOn_Ioi_comp_rpow_iff'
+-/
 
+#print MeasureTheory.integrableOn_Ioi_comp_mul_left_iff /-
 theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
     IntegrableOn (fun x => f (a * x)) (Ioi c) ↔ IntegrableOn f (Ioi <| a * c) :=
   by
@@ -1042,11 +1153,14 @@ theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (
   ext1 x
   rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c, mul_div_cancel _ ha.ne']
 #align measure_theory.integrable_on_Ioi_comp_mul_left_iff MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
+-/
 
+#print MeasureTheory.integrableOn_Ioi_comp_mul_right_iff /-
 theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
     IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) := by
   simpa only [mul_comm, MulZeroClass.mul_zero] using integrable_on_Ioi_comp_mul_left_iff f c ha
 #align measure_theory.integrable_on_Ioi_comp_mul_right_iff MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
+-/
 
 end IoiIntegrability
 

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: Anatole Dedecker, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit b84aee748341da06a6d78491367e2c0e9f15e8a5
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.MeasureTheory.Measure.Haar.NormedSpace
 /-!
 # Links between an integral and its "improper" version
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In its current state, mathlib only knows how to talk about definite ("proper") integrals,
 in the sense that it treats integrals over `[x, +∞)` the same as it treats integrals over
 `[y, z]`. For example, the integral over `[1, +∞)` is **not** defined to be the limit of

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

@@ -428,7 +428,7 @@ theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated
 
 theorem AECover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
-    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : (∫⁻ x, f x ∂μ) = I :=
+    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
 #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto
 
@@ -453,7 +453,7 @@ variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter 
 
 theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
-    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ :=
+    (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖₊ ∂μ ≤ ENNReal.ofReal I) : Integrable f μ :=
   by
   refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
   exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
@@ -472,7 +472,7 @@ theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountably
 
 theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
-    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
+    (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖₊ ∂μ ≤ I) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
@@ -486,7 +486,7 @@ theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountabl
 
 theorem AECover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
-    (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
+    (hbounded : ∀ᶠ i in l, ∫ x in φ i, ‖f x‖ ∂μ ≤ I) : Integrable f μ :=
   by
   have hfm : ae_strongly_measurable f μ :=
     hφ.ae_strongly_measurable fun i => (hfi i).AEStronglyMeasurable
@@ -512,7 +512,7 @@ theorem AECover.integrable_of_integral_norm_tendsto [l.ne_bot] [l.IsCountablyGen
 
 theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
-    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ :=
+    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, ∫ x in φ i, f x ∂μ ≤ I) : Integrable f μ :=
   hφ.integrable_of_integral_norm_bounded I hfi <|
     hbounded.mono fun i hi =>
       (integral_congr_ae <| ae_restrict_of_ae <| hnng.mono fun x => Real.norm_of_nonneg).le.trans hi
@@ -548,14 +548,14 @@ theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated]
     `measure_theory.ae_cover.integral_tendsto_of_countably_generated`. -/
 theorem AECover.integral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
-    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : (∫ x, f x ∂μ) = I :=
+    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h
 #align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto
 
 theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
     (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
-    (∫ x, f x ∂μ) = I :=
+    ∫ x, f x ∂μ = I :=
   have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto
   hφ.integral_eq_of_tendsto I hfi' htendsto
 #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
@@ -569,7 +569,7 @@ variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [Is
 
 theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
-    (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
+    (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, ∫ x in a i..b i, ‖f x‖ ∂μ ≤ I) : Integrable f μ :=
   by
   have hφ : ae_cover μ l _ := ae_cover_Ioc ha hb
   refine' hφ.integrable_of_integral_norm_bounded I hfi (h.mp _)
@@ -594,7 +594,7 @@ theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
 
 theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
-    (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ :=
+    (h : ∀ᶠ i in l, ∫ x in a i..b, ‖f x‖ ∂μ ≤ I) : IntegrableOn f (Iic b) μ :=
   by
   have hφ : ae_cover (μ.restrict <| Iic b) l _ := ae_cover_Ioi ha
   have hfi : ∀ i, integrable_on f (Ioi (a i)) (μ.restrict <| Iic b) :=
@@ -623,7 +623,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
 
 theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
-    (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ :=
+    (h : ∀ᶠ i in l, ∫ x in a..b i, ‖f x‖ ∂μ ≤ I) : IntegrableOn f (Ioi a) μ :=
   by
   have hφ : ae_cover (μ.restrict <| Ioi a) l _ := ae_cover_Iic hb
   have hfi : ∀ i, integrable_on f (Iic (b i)) (μ.restrict <| Ioi a) :=
@@ -652,7 +652,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
 
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
-    (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) :
+    (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, ∫ x in Ioc (a i) (b i), ‖f x‖ ≤ I) :
     IntegrableOn f (Ioc a₀ b₀) :=
   by
   refine'
@@ -666,13 +666,13 @@ theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
 
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
-    (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) :=
+    (h : ∀ᶠ i in l, ∫ x in Ioc (a i) b, ‖f x‖ ≤ I) : IntegrableOn f (Ioc a₀ b) :=
   integrableOn_Ioc_of_interval_integral_norm_bounded hfi ha tendsto_const_nhds h
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left
 
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded_right {I a b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
-    (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) :=
+    (h : ∀ᶠ i in l, ∫ x in Ioc a (b i), ‖f x‖ ≤ I) : IntegrableOn f (Ioc a b₀) :=
   integrableOn_Ioc_of_interval_integral_norm_bounded hfi tendsto_const_nhds hb h
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
 
@@ -741,7 +741,7 @@ integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differe
 on `(a, +∞)` and continuity on `[a, +∞)`.-/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
-    (hf : Tendsto f atTop (𝓝 m)) : (∫ x in Ioi a, f' x) = m - f a :=
+    (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a :=
   by
   refine' tendsto_nhds_unique (interval_integral_tendsto_integral_Ioi a f'int tendsto_id) _
   apply tendsto.congr' _ (hf.sub_const _)
@@ -760,8 +760,7 @@ When a function has a limit at infinity `m`, and its derivative is integrable, t
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
 on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
-    (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
-    (∫ x in Ioi a, f' x) = m - f a :=
+    (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a :=
   by
   apply integral_Ioi_of_has_deriv_at_of_tendsto _ (fun x hx => hderiv x (le_of_lt hx)) f'int hf
   intro x hx
@@ -820,7 +819,7 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 continuity on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
-    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
@@ -829,7 +828,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
-    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonneg' hderiv g'pos hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
@@ -864,7 +863,7 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 continuity on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
-    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonpos hcont hderiv g'neg hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
@@ -873,7 +872,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
-    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
@@ -896,9 +895,9 @@ theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : 
     (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
     (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
     (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) (Ici a)) :
-    (∫ x in Ioi a, f' x • (g ∘ f) x) = ∫ u in Ioi (f a), g u :=
+    ∫ x in Ioi a, f' x • (g ∘ f) x = ∫ u in Ioi (f a), g u :=
   by
-  have eq : ∀ b : ℝ, a < b → (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u :=
+  have eq : ∀ b : ℝ, a < b → ∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u :=
     by
     intro b hb
     have i1 : Ioo (min a b) (max a b) ⊆ Ioi a := by rw [min_eq_left hb.le];
@@ -928,7 +927,7 @@ theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a :
     (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
     (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
     (hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) (Ici a)) :
-    (∫ x in Ioi a, (g ∘ f) x * f' x) = ∫ u in Ioi (f a), g u :=
+    ∫ x in Ioi a, (g ∘ f) x * f' x = ∫ u in Ioi (f a), g u :=
   by
   have hg2' : integrable_on (fun x => f' x • (g ∘ f) x) (Ici a) := by simpa [mul_comm] using hg2
   simpa [mul_comm] using integral_comp_smul_deriv_Ioi hf hft hff' hg_cont hg1 hg2'
@@ -937,7 +936,7 @@ theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a :
 
 /-- Substitution `y = x ^ p` in integrals over `Ioi 0` -/
 theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
-    (∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y :=
+    ∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ y in Ioi 0, g y :=
   by
   let S := Ioi (0 : ℝ)
   have a1 : ∀ x : ℝ, x ∈ S → HasDerivWithinAt (fun t : ℝ => t ^ p) (p * x ^ (p - 1)) S x :=
@@ -964,14 +963,14 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
 #align measure_theory.integral_comp_rpow_Ioi MeasureTheory.integral_comp_rpow_Ioi
 
 theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
-    (∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y :=
+    ∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ y in Ioi 0, g y :=
   by
   convert integral_comp_rpow_Ioi g hp.ne'
   funext; congr; rw [abs_of_nonneg hp.le]
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
 
 theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
-    (∫ x in Ioi a, g (b * x)) = |b⁻¹| • ∫ x in Ioi (b * a), g x :=
+    ∫ x in Ioi a, g (b * x) = |b⁻¹| • ∫ x in Ioi (b * a), g x :=
   by
   have : ∀ c : ℝ, MeasurableSet (Ioi c) := fun c => measurableSet_Ioi
   rw [← integral_indicator (this a), ← integral_indicator (this <| b * a)]
@@ -981,7 +980,7 @@ theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 <
 #align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
 
 theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
-    (∫ x in Ioi a, g (x * b)) = |b⁻¹| • ∫ x in Ioi (a * b), g x := by
+    ∫ x in Ioi a, g (x * b) = |b⁻¹| • ∫ x in Ioi (a * b), g x := by
   simpa only [mul_comm] using integral_comp_mul_left_Ioi g a hb
 #align measure_theory.integral_comp_mul_right_Ioi MeasureTheory.integral_comp_mul_right_Ioi
 

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

@@ -79,6 +79,7 @@ section AeCover
 
 variable {α ι : Type _} [MeasurableSpace α] (μ : Measure α) (l : Filter ι)
 
+#print MeasureTheory.AECover /-
 /-- A sequence `φ` of subsets of `α` is a `ae_cover` w.r.t. a measure `μ` and a filter `l`
     if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if
     each `φ n` is measurable.
@@ -89,10 +90,11 @@ variable {α ι : Type _} [MeasurableSpace α] (μ : Measure α) (l : Filter ι)
     See for example `measure_theory.ae_cover.lintegral_tendsto_of_countably_generated`,
     `measure_theory.ae_cover.integrable_of_integral_norm_tendsto` and
     `measure_theory.ae_cover.integral_tendsto_of_countably_generated`. -/
-structure AeCover (φ : ι → Set α) : Prop where
+structure AECover (φ : ι → Set α) : Prop where
   ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
   Measurable : ∀ i, MeasurableSet <| φ i
-#align measure_theory.ae_cover MeasureTheory.AeCover
+#align measure_theory.ae_cover MeasureTheory.AECover
+-/
 
 variable {μ} {l}
 
@@ -101,25 +103,25 @@ section Preorderα
 variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
-theorem aeCover_Icc : AeCover μ l fun i => Icc (a i) (b i) :=
+theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_le_atBot x).mp <|
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Icc }
-#align measure_theory.ae_cover_Icc MeasureTheory.aeCover_Icc
+#align measure_theory.ae_cover_Icc MeasureTheory.aecover_Icc
 
-theorem aeCover_Ici : AeCover μ l fun i => Ici <| a i :=
+theorem aecover_Ici : AECover μ l fun i => Ici <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_le_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ici }
-#align measure_theory.ae_cover_Ici MeasureTheory.aeCover_Ici
+#align measure_theory.ae_cover_Ici MeasureTheory.aecover_Ici
 
-theorem aeCover_Iic : AeCover μ l fun i => Iic <| b i :=
+theorem aecover_Iic : AECover μ l fun i => Iic <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iic }
-#align measure_theory.ae_cover_Iic MeasureTheory.aeCover_Iic
+#align measure_theory.ae_cover_Iic MeasureTheory.aecover_Iic
 
 end Preorderα
 
@@ -128,41 +130,41 @@ section LinearOrderα
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
-theorem aeCover_Ioo [NoMinOrder α] [NoMaxOrder α] : AeCover μ l fun i => Ioo (a i) (b i) :=
+theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_lt_atBot x).mp <|
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioo }
-#align measure_theory.ae_cover_Ioo MeasureTheory.aeCover_Ioo
+#align measure_theory.ae_cover_Ioo MeasureTheory.aecover_Ioo
 
-theorem aeCover_Ioc [NoMinOrder α] : AeCover μ l fun i => Ioc (a i) (b i) :=
+theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_lt_atBot x).mp <|
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioc }
-#align measure_theory.ae_cover_Ioc MeasureTheory.aeCover_Ioc
+#align measure_theory.ae_cover_Ioc MeasureTheory.aecover_Ioc
 
-theorem aeCover_Ico [NoMaxOrder α] : AeCover μ l fun i => Ico (a i) (b i) :=
+theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_le_atBot x).mp <|
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ico }
-#align measure_theory.ae_cover_Ico MeasureTheory.aeCover_Ico
+#align measure_theory.ae_cover_Ico MeasureTheory.aecover_Ico
 
-theorem aeCover_Ioi [NoMinOrder α] : AeCover μ l fun i => Ioi <| a i :=
+theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_lt_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ioi }
-#align measure_theory.ae_cover_Ioi MeasureTheory.aeCover_Ioi
+#align measure_theory.ae_cover_Ioi MeasureTheory.aecover_Ioi
 
-theorem aeCover_Iio [NoMaxOrder α] : AeCover μ l fun i => Iio <| b i :=
+theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iio }
-#align measure_theory.ae_cover_Iio MeasureTheory.aeCover_Iio
+#align measure_theory.ae_cover_Iio MeasureTheory.aecover_Iio
 
 end LinearOrderα
 
@@ -171,135 +173,160 @@ section FiniteIntervals
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B))
 
-theorem aeCover_Ioo_of_Icc : AeCover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) :=
+theorem aecover_Ioo_of_Icc : AECover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_le_nhds hx.left).mp <|
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Icc }
-#align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aeCover_Ioo_of_Icc
+#align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aecover_Ioo_of_Icc
 
-theorem aeCover_Ioo_of_Ico : AeCover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) :=
+theorem aecover_Ioo_of_Ico : AECover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_le_nhds hx.left).mp <|
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ico }
-#align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aeCover_Ioo_of_Ico
+#align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aecover_Ioo_of_Ico
 
-theorem aeCover_Ioo_of_Ioc : AeCover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) :=
+theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_lt_nhds hx.left).mp <|
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioc }
-#align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aeCover_Ioo_of_Ioc
+#align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aecover_Ioo_of_Ioc
 
-theorem aeCover_Ioo_of_Ioo : AeCover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) :=
+theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_lt_nhds hx.left).mp <|
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioo }
-#align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aeCover_Ioo_of_Ioo
+#align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aecover_Ioo_of_Ioo
 
 variable [NoAtoms μ]
 
-theorem aeCover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ioc A B) l fun i => Icc (a i) (b i) := by
+#print MeasureTheory.aecover_Ioc_of_Icc /-
+theorem aecover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ioc A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aeCover_Ioc_of_Icc
+#align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aecover_Ioc_of_Icc
+-/
 
-theorem aeCover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ioc A B) l fun i => Ico (a i) (b i) := by
+#print MeasureTheory.aecover_Ioc_of_Ico /-
+theorem aecover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ioc A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aeCover_Ioc_of_Ico
+#align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aecover_Ioc_of_Ico
+-/
 
-theorem aeCover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ioc A B) l fun i => Ioc (a i) (b i) := by
+#print MeasureTheory.aecover_Ioc_of_Ioc /-
+theorem aecover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ioc A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aeCover_Ioc_of_Ioc
+#align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aecover_Ioc_of_Ioc
+-/
 
-theorem aeCover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ioc A B) l fun i => Ioo (a i) (b i) := by
+#print MeasureTheory.aecover_Ioc_of_Ioo /-
+theorem aecover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ioc A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aeCover_Ioc_of_Ioo
+#align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aecover_Ioc_of_Ioo
+-/
 
-theorem aeCover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ico A B) l fun i => Icc (a i) (b i) := by
+#print MeasureTheory.aecover_Ico_of_Icc /-
+theorem aecover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ico A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aeCover_Ico_of_Icc
+#align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aecover_Ico_of_Icc
+-/
 
-theorem aeCover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ico A B) l fun i => Ico (a i) (b i) := by
+#print MeasureTheory.aecover_Ico_of_Ico /-
+theorem aecover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ico A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aeCover_Ico_of_Ico
+#align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aecover_Ico_of_Ico
+-/
 
-theorem aeCover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ico A B) l fun i => Ioc (a i) (b i) := by
+#print MeasureTheory.aecover_Ico_of_Ioc /-
+theorem aecover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ico A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aeCover_Ico_of_Ioc
+#align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aecover_Ico_of_Ioc
+-/
 
-theorem aeCover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Ico A B) l fun i => Ioo (a i) (b i) := by
+#print MeasureTheory.aecover_Ico_of_Ioo /-
+theorem aecover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Ico A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aeCover_Ico_of_Ioo
+#align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aecover_Ico_of_Ioo
+-/
 
-theorem aeCover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Icc A B) l fun i => Icc (a i) (b i) := by
+#print MeasureTheory.aecover_Icc_of_Icc /-
+theorem aecover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Icc A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aeCover_Icc_of_Icc
+#align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aecover_Icc_of_Icc
+-/
 
-theorem aeCover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Icc A B) l fun i => Ico (a i) (b i) := by
+#print MeasureTheory.aecover_Icc_of_Ico /-
+theorem aecover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Icc A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aeCover_Icc_of_Ico
+#align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aecover_Icc_of_Ico
+-/
 
-theorem aeCover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Icc A B) l fun i => Ioc (a i) (b i) := by
+#print MeasureTheory.aecover_Icc_of_Ioc /-
+theorem aecover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Icc A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aeCover_Icc_of_Ioc
+#align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aecover_Icc_of_Ioc
+-/
 
-theorem aeCover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
-    AeCover (μ.restrict <| Icc A B) l fun i => Ioo (a i) (b i) := by
+#print MeasureTheory.aecover_Icc_of_Ioo /-
+theorem aecover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+    AECover (μ.restrict <| Icc A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aeCover_Icc_of_Ioo
+#align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aecover_Icc_of_Ioo
+-/
 
 end FiniteIntervals
 
-theorem AeCover.restrict {φ : ι → Set α} (hφ : AeCover μ l φ) {s : Set α} :
-    AeCover (μ.restrict s) l φ :=
+theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} :
+    AECover (μ.restrict s) l φ :=
   { ae_eventually_mem := ae_restrict_of_ae hφ.ae_eventually_mem
     Measurable := hφ.Measurable }
-#align measure_theory.ae_cover.restrict MeasureTheory.AeCover.restrict
+#align measure_theory.ae_cover.restrict MeasureTheory.AECover.restrict
 
-theorem aeCover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
+theorem aecover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
     (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n)
-    (measurable : ∀ n, MeasurableSet <| φ n) : AeCover (μ.restrict s) l φ :=
+    (measurable : ∀ n, MeasurableSet <| φ n) : AECover (μ.restrict s) l φ :=
   { ae_eventually_mem := by rwa [ae_restrict_iff' hs]
     Measurable }
-#align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aeCover_restrict_of_ae_imp
+#align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aecover_restrict_of_ae_imp
 
-theorem AeCover.inter_restrict {φ : ι → Set α} (hφ : AeCover μ l φ) {s : Set α}
-    (hs : MeasurableSet s) : AeCover (μ.restrict s) l fun i => φ i ∩ s :=
-  aeCover_restrict_of_ae_imp hs
+theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α}
+    (hs : MeasurableSet s) : AECover (μ.restrict s) l fun i => φ i ∩ s :=
+  aecover_restrict_of_ae_imp hs
     (hφ.ae_eventually_mem.mono fun x hx hxs => hx.mono fun i hi => ⟨hi, hxs⟩) fun i =>
     (hφ.Measurable i).inter hs
-#align measure_theory.ae_cover.inter_restrict MeasureTheory.AeCover.inter_restrict
+#align measure_theory.ae_cover.inter_restrict MeasureTheory.AECover.inter_restrict
 
-theorem AeCover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace β] (f : α → β)
-    {φ : ι → Set α} (hφ : AeCover μ l φ) :
+theorem AECover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace β] (f : α → β)
+    {φ : ι → Set α} (hφ : AECover μ l φ) :
     ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <| f x) :=
   hφ.ae_eventually_mem.mono fun x hx =>
     tendsto_const_nhds.congr' <| hx.mono fun n hn => (indicator_of_mem hn _).symm
-#align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AeCover.ae_tendsto_indicator
+#align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AECover.ae_tendsto_indicator
 
-theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.ne_bot]
-    {f : α → β} {φ : ι → Set α} (hφ : AeCover μ l φ)
+#print MeasureTheory.AECover.aemeasurable /-
+theorem AECover.aemeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.ne_bot]
+    {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
     (hfm : ∀ i, AEMeasurable f (μ.restrict <| φ i)) : AEMeasurable f μ :=
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
@@ -308,10 +335,12 @@ theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
   filter_upwards [hφ.ae_eventually_mem] with x hx using
     let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
-#align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aEMeasurable
+#align measure_theory.ae_cover.ae_measurable MeasureTheory.AECover.aemeasurable
+-/
 
-theorem AeCover.aEStronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
-    [l.IsCountablyGenerated] [l.ne_bot] {f : α → β} {φ : ι → Set α} (hφ : AeCover μ l φ)
+#print MeasureTheory.AECover.aestronglyMeasurable /-
+theorem AECover.aestronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
+    [l.IsCountablyGenerated] [l.ne_bot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
     (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <| φ i)) : AEStronglyMeasurable f μ :=
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
@@ -320,30 +349,33 @@ theorem AeCover.aEStronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
   filter_upwards [hφ.ae_eventually_mem] with x hx using
     let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
-#align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AeCover.aEStronglyMeasurable
+#align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AECover.aestronglyMeasurable
+-/
 
 end AeCover
 
-theorem AeCover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
-    {l' : Filter ι'} {φ : ι → Set α} (hφ : AeCover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
-    AeCover μ l' (φ ∘ u) :=
+theorem AECover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
+    {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
+    AECover μ l' (φ ∘ u) :=
   { ae_eventually_mem := hφ.ae_eventually_mem.mono fun x hx => hu.Eventually hx
     Measurable := fun i => hφ.Measurable (u i) }
-#align measure_theory.ae_cover.comp_tendsto MeasureTheory.AeCover.comp_tendsto
+#align measure_theory.ae_cover.comp_tendsto MeasureTheory.AECover.comp_tendsto
 
 section AeCoverUnionInterCountable
 
 variable {α ι : Type _} [Countable ι] [MeasurableSpace α] {μ : Measure α}
 
-theorem AeCover.bUnion_Iic_aeCover [Preorder ι] {φ : ι → Set α} (hφ : AeCover μ atTop φ) :
-    AeCover μ atTop fun n : ι => ⋃ (k) (h : k ∈ Iic n), φ k :=
+#print MeasureTheory.AECover.biUnion_Iic_aecover /-
+theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) :
+    AECover μ atTop fun n : ι => ⋃ (k) (h : k ∈ Iic n), φ k :=
   { ae_eventually_mem :=
       hφ.ae_eventually_mem.mono fun x h => h.mono fun i hi => mem_biUnion right_mem_Iic hi
     Measurable := fun i => MeasurableSet.biUnion (to_countable _) fun n _ => hφ.Measurable n }
-#align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AeCover.bUnion_Iic_aeCover
+#align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AECover.biUnion_Iic_aecover
+-/
 
-theorem AeCover.bInter_Ici_aeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
-    (hφ : AeCover μ atTop φ) : AeCover μ atTop fun n : ι => ⋂ (k) (h : k ∈ Ici n), φ k :=
+theorem AECover.biInter_Ici_aecover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
+    (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (h : k ∈ Ici n), φ k :=
   { ae_eventually_mem :=
       hφ.ae_eventually_mem.mono
         (by
@@ -354,7 +386,7 @@ theorem AeCover.bInter_Ici_aeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι 
           intro j hj
           exact mem_bInter fun k hk => hi k (le_trans hj hk))
     Measurable := fun i => MeasurableSet.biInter (to_countable _) fun n _ => hφ.Measurable n }
-#align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AeCover.bInter_Ici_aeCover
+#align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AECover.biInter_Ici_aecover
 
 end AeCoverUnionInterCountable
 
@@ -362,7 +394,7 @@ section Lintegral
 
 variable {α ι : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
 
-private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AeCover μ atTop φ)
+private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ)
     (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
     Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
   let F n := (φ n).indicator f
@@ -374,7 +406,7 @@ private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ
   (lintegral_tendsto_of_tendsto_of_monotone key₁ key₂ key₃).congr fun n =>
     lintegral_indicator f (hφ.Measurable n)
 
-theorem AeCover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AeCover μ atTop φ) {f : α → ℝ≥0∞}
+theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
   by
   have lim₁ :=
@@ -386,22 +418,22 @@ theorem AeCover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AeCover μ
   have le₁ := fun n => lintegral_mono_set (bInter_subset_of_mem left_mem_Ici)
   have le₂ := fun n => lintegral_mono_set (subset_bUnion_of_mem right_mem_Iic)
   exact tendsto_of_tendsto_of_tendsto_of_le_of_le lim₁ lim₂ le₁ le₂
-#align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AeCover.lintegral_tendsto_of_nat
+#align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AECover.lintegral_tendsto_of_nat
 
-theorem AeCover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
+theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
+    (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
     Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <| ∫⁻ x, f x ∂μ) :=
   tendsto_of_seq_tendsto fun u hu => (hφ.comp_tendsto hu).lintegral_tendsto_of_nat hfm
-#align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AeCover.lintegral_tendsto_of_countably_generated
+#align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AECover.lintegral_tendsto_of_countably_generated
 
-theorem AeCover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
+theorem AECover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
+    (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : (∫⁻ x, f x ∂μ) = I :=
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
-#align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AeCover.lintegral_eq_of_tendsto
+#align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto
 
-theorem AeCover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
-    [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ≥0∞}
+theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
+    [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ :=
   by
   have := hφ.lintegral_tendsto_of_countably_generated hfm
@@ -411,7 +443,7 @@ theorem AeCover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot
   rcases exists_between hw with ⟨m, hm₁, hm₂⟩
   rcases(eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
   exact ⟨i, lt_of_lt_of_le hm₁ hi⟩
-#align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AeCover.iSup_lintegral_eq_of_countably_generated
+#align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated
 
 end Lintegral
 
@@ -419,16 +451,16 @@ section Integrable
 
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
-theorem AeCover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
+theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ :=
   by
   refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
   exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded
 
-theorem AeCover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
+theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
     Integrable f μ :=
   by
@@ -436,24 +468,24 @@ theorem AeCover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountably
   refine' htendsto.eventually (ge_mem_nhds _)
   refine' (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _
   exact lt_max_of_lt_right (lt_add_one I)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_tendsto
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto
 
-theorem AeCover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
+theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded'
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
 
-theorem AeCover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
+theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_tendsto I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_tendsto'
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto'
 
-theorem AeCover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AECover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
   by
   have hfm : ae_strongly_measurable f μ :=
@@ -469,30 +501,30 @@ theorem AeCover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGen
   refine' hbounded.mono fun i hi => _
   rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)]
   apply ENNReal.ofReal_le_ofReal hi
-#align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AeCover.integrable_of_integral_norm_bounded
+#align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AECover.integrable_of_integral_norm_bounded
 
-theorem AeCover.integrable_of_integral_norm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AECover.integrable_of_integral_norm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ :=
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
   hφ.integrable_of_integral_norm_bounded I' hfi hI'
-#align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AeCover.integrable_of_integral_norm_tendsto
+#align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AECover.integrable_of_integral_norm_tendsto
 
-theorem AeCover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ :=
   hφ.integrable_of_integral_norm_bounded I hfi <|
     hbounded.mono fun i hi =>
       (integral_congr_ae <| ae_restrict_of_ae <| hnng.mono fun x => Real.norm_of_nonneg).le.trans hi
-#align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AeCover.integrable_of_integral_bounded_of_nonneg_ae
+#align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
 
-theorem AeCover.integrable_of_integral_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) :
     Integrable f μ :=
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
   hφ.integrable_of_integral_bounded_of_nonneg_ae I' hfi hnng hI'
-#align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AeCover.integrable_of_integral_tendsto_of_nonneg_ae
+#align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae
 
 end Integrable
 
@@ -501,8 +533,8 @@ section Integral
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
   [NormedSpace ℝ E] [CompleteSpace E]
 
-theorem AeCover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → E} (hfi : Integrable f μ) :
+theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
+    (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) :
     Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) :=
   suffices h (Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ))) by
     convert h; ext n; rw [integral_indicator (hφ.measurable n)]
@@ -510,23 +542,23 @@ theorem AeCover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated]
     (eventually_of_forall fun i => hfi.AEStronglyMeasurable.indicator <| hφ.Measurable i)
     (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm
     (hφ.ae_tendsto_indicator f)
-#align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AeCover.integral_tendsto_of_countably_generated
+#align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AECover.integral_tendsto_of_countably_generated
 
 /-- Slight reformulation of
     `measure_theory.ae_cover.integral_tendsto_of_countably_generated`. -/
-theorem AeCover.integral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
+theorem AECover.integral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
+    (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
     (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : (∫ x, f x ∂μ) = I :=
   tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h
-#align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AeCover.integral_eq_of_tendsto
+#align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto
 
-theorem AeCover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
+theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
     (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
     (∫ x, f x ∂μ) = I :=
   have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto
   hφ.integral_eq_of_tendsto I hfi' htendsto
-#align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AeCover.integral_eq_of_tendsto_of_nonneg_ae
+#align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
 
 end Integral
 
@@ -546,6 +578,7 @@ theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
   rwa [← intervalIntegral.integral_of_le (hai.trans hbi)]
 #align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded
 
+#print MeasureTheory.integrable_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc (a i) (b i)`,
 where `a i` tends to -∞ and `b i` tends to ∞, and
 `∫ x in a i .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
@@ -557,6 +590,7 @@ theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
   integrable_of_intervalIntegral_norm_bounded I' hfi ha hb hI'
 #align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrable_of_intervalIntegral_norm_tendsto
+-/
 
 theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
@@ -574,6 +608,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
   exact id
 #align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
 
+#print MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc (a i) b`,
 where `a i` tends to -∞, and
 `∫ x in a i .. b, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
@@ -584,6 +619,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
   integrableOn_Iic_of_intervalIntegral_norm_bounded I' b hfi ha hI'
 #align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto
+-/
 
 theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
@@ -601,6 +637,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
   exact id
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
 
+#print MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto /-
 /-- If `f` is integrable on intervals `Ioc a (b i)`,
 where `b i` tends to ∞, and
 `∫ x in a .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
@@ -611,6 +648,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
   integrableOn_Ioi_of_intervalIntegral_norm_bounded I' a hfi hb hI'
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
+-/
 
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
@@ -645,6 +683,7 @@ section IntegralOfIntervalIntegral
 variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [IsCountablyGenerated l]
   [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {a b : ι → ℝ} {f : ℝ → E}
 
+#print MeasureTheory.intervalIntegral_tendsto_integral /-
 theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot)
     (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a i..b i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) :=
   by
@@ -655,7 +694,9 @@ theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto
     hb.eventually (eventually_ge_at_top 0)] with i hai hbi
   exact (intervalIntegral.integral_of_le (hai.trans hbi)).symm
 #align measure_theory.interval_integral_tendsto_integral MeasureTheory.intervalIntegral_tendsto_integral
+-/
 
+#print MeasureTheory.intervalIntegral_tendsto_integral_Iic /-
 theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
     (ha : Tendsto a l atBot) :
     Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) :=
@@ -667,7 +708,9 @@ theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (I
   rw [intervalIntegral.integral_of_le hai, measure.restrict_restrict (hφ.measurable i)]
   rfl
 #align measure_theory.interval_integral_tendsto_integral_Iic MeasureTheory.intervalIntegral_tendsto_integral_Iic
+-/
 
+#print MeasureTheory.intervalIntegral_tendsto_integral_Ioi /-
 theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (Ioi a) μ)
     (hb : Tendsto b l atTop) :
     Tendsto (fun i => ∫ x in a..b i, f x ∂μ) l (𝓝 <| ∫ x in Ioi a, f x ∂μ) :=
@@ -679,6 +722,7 @@ theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (I
   rw [intervalIntegral.integral_of_le hbi, measure.restrict_restrict (hφ.measurable i), inter_comm]
   rfl
 #align measure_theory.interval_integral_tendsto_integral_Ioi MeasureTheory.intervalIntegral_tendsto_integral_Ioi
+-/
 
 end IntegralOfIntervalIntegral
 
@@ -844,6 +888,7 @@ open scoped Interval
 
 variable {E : Type _} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
+#print MeasureTheory.integral_comp_smul_deriv_Ioi /-
 /-- Change-of-variables formula for `Ioi` integrals of vector-valued functions, proved by taking
 limits from the result for finite intervals. -/
 theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : ℝ}
@@ -874,7 +919,9 @@ theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : 
   have t1 := (interval_integral_tendsto_integral_Ioi _ (hg1.mono_set this) tendsto_id).comp hft
   exact tendsto_nhds_unique (tendsto.congr' (eventually_eq_of_mem (Ioi_mem_at_top a) Eq) t2) t1
 #align measure_theory.integral_comp_smul_deriv_Ioi MeasureTheory.integral_comp_smul_deriv_Ioi
+-/
 
+#print MeasureTheory.integral_comp_mul_deriv_Ioi /-
 /-- Change-of-variables formula for `Ioi` integrals of scalar-valued functions -/
 theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a : ℝ}
     (hf : ContinuousOn f <| Ici a) (hft : Tendsto f atTop atTop)
@@ -886,6 +933,7 @@ theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a :
   have hg2' : integrable_on (fun x => f' x • (g ∘ f) x) (Ici a) := by simpa [mul_comm] using hg2
   simpa [mul_comm] using integral_comp_smul_deriv_Ioi hf hft hff' hg_cont hg1 hg2'
 #align measure_theory.integral_comp_mul_deriv_Ioi MeasureTheory.integral_comp_mul_deriv_Ioi
+-/
 
 /-- Substitution `y = x ^ p` in integrals over `Ioi 0` -/
 theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :

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

@@ -504,7 +504,7 @@ variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter 
 theorem AeCover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AeCover μ l φ) {f : α → E} (hfi : Integrable f μ) :
     Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) :=
-  suffices h : Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ)) from by
+  suffices h (Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ))) by
     convert h; ext n; rw [integral_indicator (hφ.measurable n)]
   tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖)
     (eventually_of_forall fun i => hfi.AEStronglyMeasurable.indicator <| hφ.Measurable i)
@@ -757,7 +757,6 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
       dsimp
       rw [abs_of_nonneg]
       exact g'pos _ hy.1
-    
 #align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
 
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative

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

@@ -305,7 +305,8 @@ theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
   rwa [measure.restrict_eq_self_of_ae_mem] at this 
-  filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
+  filter_upwards [hφ.ae_eventually_mem] with x hx using
+    let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
 #align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aEMeasurable
 
@@ -316,7 +317,8 @@ theorem AeCover.aEStronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_strongly_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
   rwa [measure.restrict_eq_self_of_ae_mem] at this 
-  filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
+  filter_upwards [hφ.ae_eventually_mem] with x hx using
+    let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
 #align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AeCover.aEStronglyMeasurable
 
@@ -540,7 +542,7 @@ theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
   have hφ : ae_cover μ l _ := ae_cover_Ioc ha hb
   refine' hφ.integrable_of_integral_norm_bounded I hfi (h.mp _)
   filter_upwards [ha.eventually (eventually_le_at_bot 0),
-    hb.eventually (eventually_ge_at_top 0)]with i hai hbi ht
+    hb.eventually (eventually_ge_at_top 0)] with i hai hbi ht
   rwa [← intervalIntegral.integral_of_le (hai.trans hbi)]
 #align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded
 
@@ -567,7 +569,7 @@ theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
     rw [integrable_on, measure.restrict_restrict (hφ.measurable i)]
     exact hfi i
   refine' hφ.integrable_of_integral_norm_bounded I hfi (h.mp _)
-  filter_upwards [ha.eventually (eventually_le_at_bot b)]with i hai
+  filter_upwards [ha.eventually (eventually_le_at_bot b)] with i hai
   rw [intervalIntegral.integral_of_le hai, measure.restrict_restrict (hφ.measurable i)]
   exact id
 #align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
@@ -594,7 +596,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
     rw [integrable_on, measure.restrict_restrict (hφ.measurable i), inter_comm]
     exact hfi i
   refine' hφ.integrable_of_integral_norm_bounded I hfi (h.mp _)
-  filter_upwards [hb.eventually (eventually_ge_at_top a)]with i hbi
+  filter_upwards [hb.eventually (eventually_ge_at_top a)] with i hbi
   rw [intervalIntegral.integral_of_le hbi, measure.restrict_restrict (hφ.measurable i), inter_comm]
   exact id
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
@@ -650,7 +652,7 @@ theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto
   have hφ : ae_cover μ l φ := ae_cover_Ioc ha hb
   refine' (hφ.integral_tendsto_of_countably_generated hfi).congr' _
   filter_upwards [ha.eventually (eventually_le_at_bot 0),
-    hb.eventually (eventually_ge_at_top 0)]with i hai hbi
+    hb.eventually (eventually_ge_at_top 0)] with i hai hbi
   exact (intervalIntegral.integral_of_le (hai.trans hbi)).symm
 #align measure_theory.interval_integral_tendsto_integral MeasureTheory.intervalIntegral_tendsto_integral
 
@@ -661,7 +663,7 @@ theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (I
   let φ i := Ioi (a i)
   have hφ : ae_cover (μ.restrict <| Iic b) l φ := ae_cover_Ioi ha
   refine' (hφ.integral_tendsto_of_countably_generated hfi).congr' _
-  filter_upwards [ha.eventually (eventually_le_at_bot <| b)]with i hai
+  filter_upwards [ha.eventually (eventually_le_at_bot <| b)] with i hai
   rw [intervalIntegral.integral_of_le hai, measure.restrict_restrict (hφ.measurable i)]
   rfl
 #align measure_theory.interval_integral_tendsto_integral_Iic MeasureTheory.intervalIntegral_tendsto_integral_Iic
@@ -673,7 +675,7 @@ theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (I
   let φ i := Iic (b i)
   have hφ : ae_cover (μ.restrict <| Ioi a) l φ := ae_cover_Iic hb
   refine' (hφ.integral_tendsto_of_countably_generated hfi).congr' _
-  filter_upwards [hb.eventually (eventually_ge_at_top <| a)]with i hbi
+  filter_upwards [hb.eventually (eventually_ge_at_top <| a)] with i hbi
   rw [intervalIntegral.integral_of_le hbi, measure.restrict_restrict (hφ.measurable i), inter_comm]
   rfl
 #align measure_theory.interval_integral_tendsto_integral_Ioi MeasureTheory.intervalIntegral_tendsto_integral_Ioi
@@ -699,7 +701,7 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
   by
   refine' tendsto_nhds_unique (interval_integral_tendsto_integral_Ioi a f'int tendsto_id) _
   apply tendsto.congr' _ (hf.sub_const _)
-  filter_upwards [Ioi_mem_at_top a]with x hx
+  filter_upwards [Ioi_mem_at_top a] with x hx
   have h'x : a ≤ id x := le_of_lt hx
   symm
   apply
@@ -736,7 +738,7 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
       intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
         (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
   apply tendsto.congr' _ (hg.sub_const _)
-  filter_upwards [Ioi_mem_at_top a]with x hx
+  filter_upwards [Ioi_mem_at_top a] with x hx
   have h'x : a ≤ id x := le_of_lt hx
   calc
     g x - g a = ∫ y in a..id x, g' y := by

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

@@ -304,7 +304,7 @@ theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
-  rwa [measure.restrict_eq_self_of_ae_mem] at this
+  rwa [measure.restrict_eq_self_of_ae_mem] at this 
   filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
 #align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aEMeasurable
@@ -315,7 +315,7 @@ theorem AeCover.aEStronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_strongly_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
-  rwa [measure.restrict_eq_self_of_ae_mem] at this
+  rwa [measure.restrict_eq_self_of_ae_mem] at this 
   filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
 #align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AeCover.aEStronglyMeasurable
@@ -864,7 +864,7 @@ theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : 
         (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2)
     · rw [min_eq_left hb.le]; exact Ioo_subset_Ioi_self
     · rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self
-  rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2
+  rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2 
   have t2 := interval_integral_tendsto_integral_Ioi _ hg2 tendsto_id
   have : Ioi (f a) ⊆ f '' Ici a :=
     Ioi_subset_Ici_self.trans <|
@@ -908,7 +908,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g
-  rw [a3] at this; rw [this]
+  rw [a3] at this ; rw [this]
   refine' set_integral_congr measurableSet_Ioi _
   intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
@@ -918,7 +918,7 @@ theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
     (∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y :=
   by
   convert integral_comp_rpow_Ioi g hp.ne'
-  funext; congr ; rw [abs_of_nonneg hp.le]
+  funext; congr; rw [abs_of_nonneg hp.le]
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
 
 theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
@@ -968,7 +968,7 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
     · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integrable_on_image_iff_integrable_on_abs_deriv_smul measurableSet_Ioi a1 a2 f
-  rw [a3] at this
+  rw [a3] at this 
   rw [this]
   refine' integrable_on_congr_fun (fun x hx => _) measurableSet_Ioi
   simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]

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

@@ -71,7 +71,7 @@ in analysis. In particular,
 
 open MeasureTheory Filter Set TopologicalSpace
 
-open ENNReal NNReal Topology
+open scoped ENNReal NNReal Topology
 
 namespace MeasureTheory
 
@@ -682,7 +682,7 @@ end IntegralOfIntervalIntegral
 
 open Real
 
-open Interval
+open scoped Interval
 
 section IoiFTC
 
@@ -839,7 +839,7 @@ section IoiChangeVariables
 
 open Real
 
-open Interval
+open scoped Interval
 
 variable {E : Type _} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
@@ -942,7 +942,7 @@ section IoiIntegrability
 
 open Real
 
-open Interval
+open scoped Interval
 
 variable {E : Type _} [NormedAddCommGroup E]
 

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

@@ -502,11 +502,8 @@ variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter 
 theorem AeCover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AeCover μ l φ) {f : α → E} (hfi : Integrable f μ) :
     Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) :=
-  suffices h : Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ)) from
-    by
-    convert h
-    ext n
-    rw [integral_indicator (hφ.measurable n)]
+  suffices h : Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ)) from by
+    convert h; ext n; rw [integral_indicator (hφ.measurable n)]
   tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖)
     (eventually_of_forall fun i => hfi.AEStronglyMeasurable.indicator <| hφ.Measurable i)
     (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm
@@ -623,11 +620,8 @@ theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
       (fun i => (hfi i).restrict measurableSet_Ioc) (eventually.mono h _)
   intro i hi; simp only [measure.restrict_restrict measurableSet_Ioc]
   refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi
-  · apply ae_of_all
-    simp only [Pi.zero_apply, norm_nonneg, forall_const]
-  · apply ae_of_all
-    intro c hc
-    exact hc.1
+  · apply ae_of_all; simp only [Pi.zero_apply, norm_nonneg, forall_const]
+  · apply ae_of_all; intro c hc; exact hc.1
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded
 
 theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
@@ -861,21 +855,15 @@ theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : 
   have eq : ∀ b : ℝ, a < b → (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u :=
     by
     intro b hb
-    have i1 : Ioo (min a b) (max a b) ⊆ Ioi a :=
-      by
-      rw [min_eq_left hb.le]
+    have i1 : Ioo (min a b) (max a b) ⊆ Ioi a := by rw [min_eq_left hb.le];
       exact Ioo_subset_Ioi_self
-    have i2 : [a, b] ⊆ Ici a := by
-      rw [uIcc_of_le hb.le]
-      exact Icc_subset_Ici_self
+    have i2 : [a, b] ⊆ Ici a := by rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self
     refine'
       intervalIntegral.integral_comp_smul_deriv''' (hf.mono i2)
         (fun x hx => hff' x <| mem_of_mem_of_subset hx i1) (hg_cont.mono <| image_subset _ _)
         (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2)
-    · rw [min_eq_left hb.le]
-      exact Ioo_subset_Ioi_self
-    · rw [uIcc_of_le hb.le]
-      exact Icc_subset_Ici_self
+    · rw [min_eq_left hb.le]; exact Ioo_subset_Ioi_self
+    · rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self
   rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2
   have t2 := interval_integral_tendsto_integral_Ioi _ hg2 tendsto_id
   have : Ioi (f a) ⊆ f '' Ici a :=
@@ -915,20 +903,14 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
       exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)
     exact StrictMonoOn.injOn fun x hx y hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h
   have a3 : (fun t : ℝ => t ^ p) '' S = S := by
-    ext1
-    rw [mem_image]
-    constructor
-    · rintro ⟨y, hy, rfl⟩
-      exact rpow_pos_of_pos hy p
-    · intro hx
-      refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
+    ext1; rw [mem_image]; constructor
+    · rintro ⟨y, hy, rfl⟩; exact rpow_pos_of_pos hy p
+    · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g
-  rw [a3] at this
-  rw [this]
+  rw [a3] at this; rw [this]
   refine' set_integral_congr measurableSet_Ioi _
-  intro x hx
-  dsimp only
+  intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
 #align measure_theory.integral_comp_rpow_Ioi MeasureTheory.integral_comp_rpow_Ioi
 
@@ -981,13 +963,9 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
       exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)
     exact StrictMonoOn.injOn fun x hx y hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h
   have a3 : (fun t : ℝ => t ^ p) '' S = S := by
-    ext1
-    rw [mem_image]
-    constructor
-    · rintro ⟨y, hy, rfl⟩
-      exact rpow_pos_of_pos hy p
-    · intro hx
-      refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
+    ext1; rw [mem_image]; constructor
+    · rintro ⟨y, hy, rfl⟩; exact rpow_pos_of_pos hy p
+    · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integrable_on_image_iff_integrable_on_abs_deriv_smul measurableSet_Ioi a1 a2 f
   rw [a3] at this

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
+! leanprover-community/mathlib commit b84aee748341da06a6d78491367e2c0e9f15e8a5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,6 +12,7 @@ import Mathbin.Analysis.SpecialFunctions.Pow.Deriv
 import Mathbin.MeasureTheory.Integral.FundThmCalculus
 import Mathbin.Order.Filter.AtTopBot
 import Mathbin.MeasureTheory.Function.Jacobian
+import Mathbin.MeasureTheory.Measure.Haar.NormedSpace
 
 /-!
 # Links between an integral and its "improper" version
@@ -370,7 +371,6 @@ private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ
   have key₃ : ∀ᵐ x : α ∂μ, Tendsto (fun n => F n x) atTop (𝓝 (f x)) := hφ.ae_tendsto_indicator f
   (lintegral_tendsto_of_tendsto_of_monotone key₁ key₂ key₃).congr fun n =>
     lintegral_indicator f (hφ.Measurable n)
-#align measure_theory.lintegral_tendsto_of_monotone_of_nat measure_theory.lintegral_tendsto_of_monotone_of_nat
 
 theorem AeCover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AeCover μ atTop φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
@@ -939,7 +939,87 @@ theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
   funext; congr ; rw [abs_of_nonneg hp.le]
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
 
+theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
+    (∫ x in Ioi a, g (b * x)) = |b⁻¹| • ∫ x in Ioi (b * a), g x :=
+  by
+  have : ∀ c : ℝ, MeasurableSet (Ioi c) := fun c => measurableSet_Ioi
+  rw [← integral_indicator (this a), ← integral_indicator (this <| b * a)]
+  convert measure.integral_comp_mul_left _ b
+  ext1 x
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left _ hb.ne']
+#align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
+
+theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
+    (∫ x in Ioi a, g (x * b)) = |b⁻¹| • ∫ x in Ioi (a * b), g x := by
+  simpa only [mul_comm] using integral_comp_mul_left_Ioi g a hb
+#align measure_theory.integral_comp_mul_right_Ioi MeasureTheory.integral_comp_mul_right_Ioi
+
 end IoiChangeVariables
 
+section IoiIntegrability
+
+open Real
+
+open Interval
+
+variable {E : Type _} [NormedAddCommGroup E]
+
+/-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability. -/
+theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
+    IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) :=
+  by
+  let S := Ioi (0 : ℝ)
+  have a1 : ∀ x : ℝ, x ∈ S → HasDerivWithinAt (fun t : ℝ => t ^ p) (p * x ^ (p - 1)) S x :=
+    fun x hx => (has_deriv_at_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).HasDerivWithinAt
+  have a2 : inj_on (fun x : ℝ => x ^ p) S :=
+    by
+    rcases lt_or_gt_of_ne hp with ⟨⟩
+    · apply StrictAntiOn.injOn
+      intro x hx y hy hxy
+      rw [← inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx), ←
+        rpow_neg (le_of_lt hy)]
+      exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)
+    exact StrictMonoOn.injOn fun x hx y hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h
+  have a3 : (fun t : ℝ => t ^ p) '' S = S := by
+    ext1
+    rw [mem_image]
+    constructor
+    · rintro ⟨y, hy, rfl⟩
+      exact rpow_pos_of_pos hy p
+    · intro hx
+      refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
+      rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
+  have := integrable_on_image_iff_integrable_on_abs_deriv_smul measurableSet_Ioi a1 a2 f
+  rw [a3] at this
+  rw [this]
+  refine' integrable_on_congr_fun (fun x hx => _) measurableSet_Ioi
+  simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
+#align measure_theory.integrable_on_Ioi_comp_rpow_iff MeasureTheory.integrableOn_Ioi_comp_rpow_iff
+
+/-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability (version
+without `|p|` factor) -/
+theorem integrableOn_Ioi_comp_rpow_iff' [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
+    IntegrableOn (fun x => x ^ (p - 1) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) := by
+  simpa only [← integrable_on_Ioi_comp_rpow_iff f hp, mul_smul] using
+    (integrable_smul_iff (abs_pos.mpr hp).ne' _).symm
+#align measure_theory.integrable_on_Ioi_comp_rpow_iff' MeasureTheory.integrableOn_Ioi_comp_rpow_iff'
+
+theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
+    IntegrableOn (fun x => f (a * x)) (Ioi c) ↔ IntegrableOn f (Ioi <| a * c) :=
+  by
+  rw [← integrable_indicator_iff (measurableSet_Ioi : MeasurableSet <| Ioi c)]
+  rw [← integrable_indicator_iff (measurableSet_Ioi : MeasurableSet <| Ioi <| a * c)]
+  convert integrable_comp_mul_left_iff ((Ioi (a * c)).indicator f) ha.ne' using 2
+  ext1 x
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c, mul_div_cancel _ ha.ne']
+#align measure_theory.integrable_on_Ioi_comp_mul_left_iff MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
+
+theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
+    IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) := by
+  simpa only [mul_comm, MulZeroClass.mul_zero] using integrable_on_Ioi_comp_mul_left_iff f c ha
+#align measure_theory.integrable_on_Ioi_comp_mul_right_iff MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
+
+end IoiIntegrability
+
 end MeasureTheory
 

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

@@ -308,16 +308,16 @@ theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
     mem_Union.mpr ⟨i, hi⟩
 #align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aEMeasurable
 
-theorem AeCover.aeStronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
+theorem AeCover.aEStronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
     [l.IsCountablyGenerated] [l.ne_bot] {f : α → β} {φ : ι → Set α} (hφ : AeCover μ l φ)
-    (hfm : ∀ i, AeStronglyMeasurable f (μ.restrict <| φ i)) : AeStronglyMeasurable f μ :=
+    (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <| φ i)) : AEStronglyMeasurable f μ :=
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_strongly_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
   rwa [measure.restrict_eq_self_of_ae_mem] at this
   filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
-#align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AeCover.aeStronglyMeasurable
+#align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AeCover.aEStronglyMeasurable
 
 end AeCover
 
@@ -418,7 +418,7 @@ section Integrable
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
 theorem AeCover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ :=
   by
   refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
@@ -426,7 +426,7 @@ theorem AeCover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountably
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded
 
 theorem AeCover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
     Integrable f μ :=
   by
@@ -437,14 +437,14 @@ theorem AeCover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountably
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_tendsto
 
 theorem AeCover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded'
 
 theorem AeCover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountablyGenerated]
-    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
   hφ.integrable_of_lintegral_nnnorm_tendsto I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
@@ -455,7 +455,7 @@ theorem AeCover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGen
     (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
   by
   have hfm : ae_strongly_measurable f μ :=
-    hφ.ae_strongly_measurable fun i => (hfi i).AeStronglyMeasurable
+    hφ.ae_strongly_measurable fun i => (hfi i).AEStronglyMeasurable
   refine' hφ.integrable_of_lintegral_nnnorm_bounded I hfm _
   conv at hbounded in integral _ _ =>
     rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x))
@@ -508,7 +508,7 @@ theorem AeCover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated]
     ext n
     rw [integral_indicator (hφ.measurable n)]
   tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖)
-    (eventually_of_forall fun i => hfi.AeStronglyMeasurable.indicator <| hφ.Measurable i)
+    (eventually_of_forall fun i => hfi.AEStronglyMeasurable.indicator <| hφ.Measurable i)
     (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm
     (hφ.ae_tendsto_indicator f)
 #align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AeCover.integral_tendsto_of_countably_generated

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit 011cafb4a5bc695875d186e245d6b3df03bf6c40
+! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Analysis.SpecialFunctions.Pow.Deriv
 import Mathbin.MeasureTheory.Integral.FundThmCalculus
 import Mathbin.Order.Filter.AtTopBot
 import Mathbin.MeasureTheory.Function.Jacobian

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

@@ -334,8 +334,8 @@ variable {α ι : Type _} [Countable ι] [MeasurableSpace α] {μ : Measure α}
 theorem AeCover.bUnion_Iic_aeCover [Preorder ι] {φ : ι → Set α} (hφ : AeCover μ atTop φ) :
     AeCover μ atTop fun n : ι => ⋃ (k) (h : k ∈ Iic n), φ k :=
   { ae_eventually_mem :=
-      hφ.ae_eventually_mem.mono fun x h => h.mono fun i hi => mem_bunionᵢ right_mem_Iic hi
-    Measurable := fun i => MeasurableSet.bunionᵢ (to_countable _) fun n _ => hφ.Measurable n }
+      hφ.ae_eventually_mem.mono fun x h => h.mono fun i hi => mem_biUnion right_mem_Iic hi
+    Measurable := fun i => MeasurableSet.biUnion (to_countable _) fun n _ => hφ.Measurable n }
 #align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AeCover.bUnion_Iic_aeCover
 
 theorem AeCover.bInter_Ici_aeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
@@ -349,7 +349,7 @@ theorem AeCover.bInter_Ici_aeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι 
           use i
           intro j hj
           exact mem_bInter fun k hk => hi k (le_trans hj hk))
-    Measurable := fun i => MeasurableSet.binterᵢ (to_countable _) fun n _ => hφ.Measurable n }
+    Measurable := fun i => MeasurableSet.biInter (to_countable _) fun n _ => hφ.Measurable n }
 #align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AeCover.bInter_Ici_aeCover
 
 end AeCoverUnionInterCountable
@@ -397,18 +397,18 @@ theorem AeCover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
 #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AeCover.lintegral_eq_of_tendsto
 
-theorem AeCover.supᵢ_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
+theorem AeCover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
     [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ≥0∞}
     (hfm : AEMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ :=
   by
   have := hφ.lintegral_tendsto_of_countably_generated hfm
   refine'
-    csupᵢ_eq_of_forall_le_of_forall_lt_exists_gt
+    ciSup_eq_of_forall_le_of_forall_lt_exists_gt
       (fun i => lintegral_mono' measure.restrict_le_self le_rfl) fun w hw => _
   rcases exists_between hw with ⟨m, hm₁, hm₂⟩
   rcases(eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
   exact ⟨i, lt_of_lt_of_le hm₁ hi⟩
-#align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AeCover.supᵢ_lintegral_eq_of_countably_generated
+#align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AeCover.iSup_lintegral_eq_of_countably_generated
 
 end Lintegral
 

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

@@ -205,7 +205,7 @@ theorem aeCover_Ioo_of_Ioo : AeCover (μ.restrict <| Ioo A B) l fun i => Ioo (a
     Measurable := fun i => measurableSet_Ioo }
 #align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aeCover_Ioo_of_Ioo
 
-variable [HasNoAtoms μ]
+variable [NoAtoms μ]
 
 theorem aeCover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ioc A B) l fun i => Icc (a i) (b i) := by

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

@@ -99,25 +99,25 @@ section Preorderα
 variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
-theorem aeCoverIcc : AeCover μ l fun i => Icc (a i) (b i) :=
+theorem aeCover_Icc : AeCover μ l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_le_atBot x).mp <|
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Icc }
-#align measure_theory.ae_cover_Icc MeasureTheory.aeCoverIcc
+#align measure_theory.ae_cover_Icc MeasureTheory.aeCover_Icc
 
-theorem aeCoverIci : AeCover μ l fun i => Ici <| a i :=
+theorem aeCover_Ici : AeCover μ l fun i => Ici <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_le_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ici }
-#align measure_theory.ae_cover_Ici MeasureTheory.aeCoverIci
+#align measure_theory.ae_cover_Ici MeasureTheory.aeCover_Ici
 
-theorem aeCoverIic : AeCover μ l fun i => Iic <| b i :=
+theorem aeCover_Iic : AeCover μ l fun i => Iic <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iic }
-#align measure_theory.ae_cover_Iic MeasureTheory.aeCoverIic
+#align measure_theory.ae_cover_Iic MeasureTheory.aeCover_Iic
 
 end Preorderα
 
@@ -126,41 +126,41 @@ section LinearOrderα
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop)
 
-theorem aeCoverIoo [NoMinOrder α] [NoMaxOrder α] : AeCover μ l fun i => Ioo (a i) (b i) :=
+theorem aeCover_Ioo [NoMinOrder α] [NoMaxOrder α] : AeCover μ l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_lt_atBot x).mp <|
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioo }
-#align measure_theory.ae_cover_Ioo MeasureTheory.aeCoverIoo
+#align measure_theory.ae_cover_Ioo MeasureTheory.aeCover_Ioo
 
-theorem aeCoverIoc [NoMinOrder α] : AeCover μ l fun i => Ioc (a i) (b i) :=
+theorem aeCover_Ioc [NoMinOrder α] : AeCover μ l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_lt_atBot x).mp <|
           (hb.Eventually <| eventually_ge_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ioc }
-#align measure_theory.ae_cover_Ioc MeasureTheory.aeCoverIoc
+#align measure_theory.ae_cover_Ioc MeasureTheory.aeCover_Ioc
 
-theorem aeCoverIco [NoMaxOrder α] : AeCover μ l fun i => Ico (a i) (b i) :=
+theorem aeCover_Ico [NoMaxOrder α] : AeCover μ l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       ae_of_all μ fun x =>
         (ha.Eventually <| eventually_le_atBot x).mp <|
           (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi hai => ⟨hai, hbi⟩
     Measurable := fun i => measurableSet_Ico }
-#align measure_theory.ae_cover_Ico MeasureTheory.aeCoverIco
+#align measure_theory.ae_cover_Ico MeasureTheory.aeCover_Ico
 
-theorem aeCoverIoi [NoMinOrder α] : AeCover μ l fun i => Ioi <| a i :=
+theorem aeCover_Ioi [NoMinOrder α] : AeCover μ l fun i => Ioi <| a i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (ha.Eventually <| eventually_lt_atBot x).mono fun i hai => hai
     Measurable := fun i => measurableSet_Ioi }
-#align measure_theory.ae_cover_Ioi MeasureTheory.aeCoverIoi
+#align measure_theory.ae_cover_Ioi MeasureTheory.aeCover_Ioi
 
-theorem aeCoverIio [NoMaxOrder α] : AeCover μ l fun i => Iio <| b i :=
+theorem aeCover_Iio [NoMaxOrder α] : AeCover μ l fun i => Iio <| b i :=
   { ae_eventually_mem :=
       ae_of_all μ fun x => (hb.Eventually <| eventually_gt_atTop x).mono fun i hbi => hbi
     Measurable := fun i => measurableSet_Iio }
-#align measure_theory.ae_cover_Iio MeasureTheory.aeCoverIio
+#align measure_theory.ae_cover_Iio MeasureTheory.aeCover_Iio
 
 end LinearOrderα
 
@@ -169,103 +169,103 @@ section FiniteIntervals
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B))
 
-theorem aeCoverIooOfIcc : AeCover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) :=
+theorem aeCover_Ioo_of_Icc : AeCover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_le_nhds hx.left).mp <|
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Icc }
-#align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aeCoverIooOfIcc
+#align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aeCover_Ioo_of_Icc
 
-theorem aeCoverIooOfIco : AeCover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) :=
+theorem aeCover_Ioo_of_Ico : AeCover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_le_nhds hx.left).mp <|
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ico }
-#align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aeCoverIooOfIco
+#align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aeCover_Ioo_of_Ico
 
-theorem aeCoverIooOfIoc : AeCover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) :=
+theorem aeCover_Ioo_of_Ioc : AeCover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_lt_nhds hx.left).mp <|
             (hb.Eventually <| eventually_ge_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioc }
-#align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aeCoverIooOfIoc
+#align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aeCover_Ioo_of_Ioc
 
-theorem aeCoverIooOfIoo : AeCover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) :=
+theorem aeCover_Ioo_of_Ioo : AeCover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) :=
   { ae_eventually_mem :=
       (ae_restrict_iff' measurableSet_Ioo).mpr
         (ae_of_all μ fun x hx =>
           (ha.Eventually <| eventually_lt_nhds hx.left).mp <|
             (hb.Eventually <| eventually_gt_nhds hx.right).mono fun i hbi hai => ⟨hai, hbi⟩)
     Measurable := fun i => measurableSet_Ioo }
-#align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aeCoverIooOfIoo
+#align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aeCover_Ioo_of_Ioo
 
 variable [HasNoAtoms μ]
 
-theorem aeCoverIocOfIcc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ioc A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aeCoverIocOfIcc
+#align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aeCover_Ioc_of_Icc
 
-theorem aeCoverIocOfIco (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ioc A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aeCoverIocOfIco
+#align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aeCover_Ioc_of_Ico
 
-theorem aeCoverIocOfIoc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ioc A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aeCoverIocOfIoc
+#align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aeCover_Ioc_of_Ioc
 
-theorem aeCoverIocOfIoo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ioc A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ioc.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aeCoverIocOfIoo
+#align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aeCover_Ioc_of_Ioo
 
-theorem aeCoverIcoOfIcc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ico A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aeCoverIcoOfIcc
+#align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aeCover_Ico_of_Icc
 
-theorem aeCoverIcoOfIco (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ico A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aeCoverIcoOfIco
+#align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aeCover_Ico_of_Ico
 
-theorem aeCoverIcoOfIoc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ico A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aeCoverIcoOfIoc
+#align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aeCover_Ico_of_Ioc
 
-theorem aeCoverIcoOfIoo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Ico A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Ico.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aeCoverIcoOfIoo
+#align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aeCover_Ico_of_Ioo
 
-theorem aeCoverIccOfIcc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Icc A B) l fun i => Icc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Icc ha hb]
-#align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aeCoverIccOfIcc
+#align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aeCover_Icc_of_Icc
 
-theorem aeCoverIccOfIco (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Icc A B) l fun i => Ico (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ico ha hb]
-#align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aeCoverIccOfIco
+#align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aeCover_Icc_of_Ico
 
-theorem aeCoverIccOfIoc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Icc A B) l fun i => Ioc (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ioc ha hb]
-#align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aeCoverIccOfIoc
+#align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aeCover_Icc_of_Ioc
 
-theorem aeCoverIccOfIoo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
+theorem aeCover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
     AeCover (μ.restrict <| Icc A B) l fun i => Ioo (a i) (b i) := by
   simp [measure.restrict_congr_set Ioo_ae_eq_Icc.symm, ae_cover_Ioo_of_Ioo ha hb]
-#align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aeCoverIccOfIoo
+#align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aeCover_Icc_of_Ioo
 
 end FiniteIntervals
 
@@ -275,19 +275,19 @@ theorem AeCover.restrict {φ : ι → Set α} (hφ : AeCover μ l φ) {s : Set 
     Measurable := hφ.Measurable }
 #align measure_theory.ae_cover.restrict MeasureTheory.AeCover.restrict
 
-theorem aeCoverRestrictOfAeImp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
+theorem aeCover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
     (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n)
     (measurable : ∀ n, MeasurableSet <| φ n) : AeCover (μ.restrict s) l φ :=
   { ae_eventually_mem := by rwa [ae_restrict_iff' hs]
     Measurable }
-#align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aeCoverRestrictOfAeImp
+#align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aeCover_restrict_of_ae_imp
 
-theorem AeCover.interRestrict {φ : ι → Set α} (hφ : AeCover μ l φ) {s : Set α}
+theorem AeCover.inter_restrict {φ : ι → Set α} (hφ : AeCover μ l φ) {s : Set α}
     (hs : MeasurableSet s) : AeCover (μ.restrict s) l fun i => φ i ∩ s :=
-  aeCoverRestrictOfAeImp hs
+  aeCover_restrict_of_ae_imp hs
     (hφ.ae_eventually_mem.mono fun x hx hxs => hx.mono fun i hi => ⟨hi, hxs⟩) fun i =>
     (hφ.Measurable i).inter hs
-#align measure_theory.ae_cover.inter_restrict MeasureTheory.AeCover.interRestrict
+#align measure_theory.ae_cover.inter_restrict MeasureTheory.AeCover.inter_restrict
 
 theorem AeCover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace β] (f : α → β)
     {φ : ι → Set α} (hφ : AeCover μ l φ) :
@@ -296,16 +296,16 @@ theorem AeCover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace 
     tendsto_const_nhds.congr' <| hx.mono fun n hn => (indicator_of_mem hn _).symm
 #align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AeCover.ae_tendsto_indicator
 
-theorem AeCover.aeMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.ne_bot]
+theorem AeCover.aEMeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.ne_bot]
     {f : α → β} {φ : ι → Set α} (hφ : AeCover μ l φ)
-    (hfm : ∀ i, AeMeasurable f (μ.restrict <| φ i)) : AeMeasurable f μ :=
+    (hfm : ∀ i, AEMeasurable f (μ.restrict <| φ i)) : AEMeasurable f μ :=
   by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
   have := ae_measurable_Union_iff.mpr fun n : ℕ => hfm (u n)
   rwa [measure.restrict_eq_self_of_ae_mem] at this
   filter_upwards [hφ.ae_eventually_mem]with x hx using let ⟨i, hi⟩ := (hu.eventually hx).exists
     mem_Union.mpr ⟨i, hi⟩
-#align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aeMeasurable
+#align measure_theory.ae_cover.ae_measurable MeasureTheory.AeCover.aEMeasurable
 
 theorem AeCover.aeStronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
     [l.IsCountablyGenerated] [l.ne_bot] {f : α → β} {φ : ι → Set α} (hφ : AeCover μ l φ)
@@ -320,25 +320,25 @@ theorem AeCover.aeStronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
 
 end AeCover
 
-theorem AeCover.compTendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
+theorem AeCover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
     {l' : Filter ι'} {φ : ι → Set α} (hφ : AeCover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
     AeCover μ l' (φ ∘ u) :=
   { ae_eventually_mem := hφ.ae_eventually_mem.mono fun x hx => hu.Eventually hx
     Measurable := fun i => hφ.Measurable (u i) }
-#align measure_theory.ae_cover.comp_tendsto MeasureTheory.AeCover.compTendsto
+#align measure_theory.ae_cover.comp_tendsto MeasureTheory.AeCover.comp_tendsto
 
 section AeCoverUnionInterCountable
 
 variable {α ι : Type _} [Countable ι] [MeasurableSpace α] {μ : Measure α}
 
-theorem AeCover.bUnionIicAeCover [Preorder ι] {φ : ι → Set α} (hφ : AeCover μ atTop φ) :
+theorem AeCover.bUnion_Iic_aeCover [Preorder ι] {φ : ι → Set α} (hφ : AeCover μ atTop φ) :
     AeCover μ atTop fun n : ι => ⋃ (k) (h : k ∈ Iic n), φ k :=
   { ae_eventually_mem :=
       hφ.ae_eventually_mem.mono fun x h => h.mono fun i hi => mem_bunionᵢ right_mem_Iic hi
     Measurable := fun i => MeasurableSet.bunionᵢ (to_countable _) fun n _ => hφ.Measurable n }
-#align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AeCover.bUnionIicAeCover
+#align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AeCover.bUnion_Iic_aeCover
 
-theorem AeCover.bInterIciAeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
+theorem AeCover.bInter_Ici_aeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι → Set α}
     (hφ : AeCover μ atTop φ) : AeCover μ atTop fun n : ι => ⋂ (k) (h : k ∈ Ici n), φ k :=
   { ae_eventually_mem :=
       hφ.ae_eventually_mem.mono
@@ -350,7 +350,7 @@ theorem AeCover.bInterIciAeCover [SemilatticeSup ι] [Nonempty ι] {φ : ι →
           intro j hj
           exact mem_bInter fun k hk => hi k (le_trans hj hk))
     Measurable := fun i => MeasurableSet.binterᵢ (to_countable _) fun n _ => hφ.Measurable n }
-#align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AeCover.bInterIciAeCover
+#align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AeCover.bInter_Ici_aeCover
 
 end AeCoverUnionInterCountable
 
@@ -359,10 +359,10 @@ section Lintegral
 variable {α ι : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
 
 private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AeCover μ atTop φ)
-    (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AeMeasurable f μ) :
+    (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
     Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
   let F n := (φ n).indicator f
-  have key₁ : ∀ n, AeMeasurable (F n) μ := fun n => hfm.indicator (hφ.Measurable n)
+  have key₁ : ∀ n, AEMeasurable (F n) μ := fun n => hfm.indicator (hφ.Measurable n)
   have key₂ : ∀ᵐ x : α ∂μ, Monotone fun n => F n x :=
     ae_of_all _ fun x i j hij =>
       indicator_le_indicator_of_subset (hmono hij) (fun x => zero_le <| f x) x
@@ -372,7 +372,7 @@ private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ
 #align measure_theory.lintegral_tendsto_of_monotone_of_nat measure_theory.lintegral_tendsto_of_monotone_of_nat
 
 theorem AeCover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AeCover μ atTop φ) {f : α → ℝ≥0∞}
-    (hfm : AeMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
+    (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) :=
   by
   have lim₁ :=
     lintegral_tendsto_of_monotone_of_nat hφ.bInter_Ici_ae_cover
@@ -386,20 +386,20 @@ theorem AeCover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AeCover μ
 #align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AeCover.lintegral_tendsto_of_nat
 
 theorem AeCover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (hfm : AeMeasurable f μ) :
+    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
     Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <| ∫⁻ x, f x ∂μ) :=
   tendsto_of_seq_tendsto fun u hu => (hφ.comp_tendsto hu).lintegral_tendsto_of_nat hfm
 #align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AeCover.lintegral_tendsto_of_countably_generated
 
 theorem AeCover.lintegral_eq_of_tendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AeMeasurable f μ)
+    (hφ : AeCover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : (∫⁻ x, f x ∂μ) = I :=
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
 #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AeCover.lintegral_eq_of_tendsto
 
 theorem AeCover.supᵢ_lintegral_eq_of_countably_generated [Nonempty ι] [l.ne_bot]
     [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ≥0∞}
-    (hfm : AeMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ :=
+    (hfm : AEMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ :=
   by
   have := hφ.lintegral_tendsto_of_countably_generated hfm
   refine'
@@ -416,15 +416,15 @@ section Integrable
 
 variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
-theorem AeCover.integrableOfLintegralNnnormBounded [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_lintegral_nnnorm_bounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ :=
   by
   refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
   exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AeCover.integrableOfLintegralNnnormBounded
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded
 
-theorem AeCover.integrableOfLintegralNnnormTendsto [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_lintegral_nnnorm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
     Integrable f μ :=
@@ -433,24 +433,24 @@ theorem AeCover.integrableOfLintegralNnnormTendsto [l.ne_bot] [l.IsCountablyGene
   refine' htendsto.eventually (ge_mem_nhds _)
   refine' (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _
   exact lt_max_of_lt_right (lt_add_one I)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AeCover.integrableOfLintegralNnnormTendsto
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_tendsto
 
-theorem AeCover.integrableOfLintegralNnnormBounded' [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_lintegral_nnnorm_bounded' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
-  hφ.integrableOfLintegralNnnormBounded I hfm
+  hφ.integrable_of_lintegral_nnnorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AeCover.integrableOfLintegralNnnormBounded'
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_bounded'
 
-theorem AeCover.integrableOfLintegralNnnormTendsto' [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_lintegral_nnnorm_tendsto' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
-  hφ.integrableOfLintegralNnnormTendsto I hfm
+  hφ.integrable_of_lintegral_nnnorm_tendsto I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
-#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AeCover.integrableOfLintegralNnnormTendsto'
+#align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AeCover.integrable_of_lintegral_nnnorm_tendsto'
 
-theorem AeCover.integrableOfIntegralNormBounded [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AeCover.integrable_of_integral_norm_bounded [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
   by
   have hfm : ae_strongly_measurable f μ :=
@@ -466,30 +466,30 @@ theorem AeCover.integrableOfIntegralNormBounded [l.ne_bot] [l.IsCountablyGenerat
   refine' hbounded.mono fun i hi => _
   rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)]
   apply ENNReal.ofReal_le_ofReal hi
-#align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AeCover.integrableOfIntegralNormBounded
+#align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AeCover.integrable_of_integral_norm_bounded
 
-theorem AeCover.integrableOfIntegralNormTendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
-    (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
+theorem AeCover.integrable_of_integral_norm_tendsto [l.ne_bot] [l.IsCountablyGenerated]
+    {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ :=
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
-  hφ.integrableOfIntegralNormBounded I' hfi hI'
-#align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AeCover.integrableOfIntegralNormTendsto
+  hφ.integrable_of_integral_norm_bounded I' hfi hI'
+#align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AeCover.integrable_of_integral_norm_tendsto
 
-theorem AeCover.integrableOfIntegralBoundedOfNonnegAe [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_integral_bounded_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ :=
-  hφ.integrableOfIntegralNormBounded I hfi <|
+  hφ.integrable_of_integral_norm_bounded I hfi <|
     hbounded.mono fun i hi =>
       (integral_congr_ae <| ae_restrict_of_ae <| hnng.mono fun x => Real.norm_of_nonneg).le.trans hi
-#align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AeCover.integrableOfIntegralBoundedOfNonnegAe
+#align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AeCover.integrable_of_integral_bounded_of_nonneg_ae
 
-theorem AeCover.integrableOfIntegralTendstoOfNonnegAe [l.ne_bot] [l.IsCountablyGenerated]
+theorem AeCover.integrable_of_integral_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) :
     Integrable f μ :=
   let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le
-  hφ.integrableOfIntegralBoundedOfNonnegAe I' hfi hnng hI'
-#align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AeCover.integrableOfIntegralTendstoOfNonnegAe
+  hφ.integrable_of_integral_bounded_of_nonneg_ae I' hfi hnng hI'
+#align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AeCover.integrable_of_integral_tendsto_of_nonneg_ae
 
 end Integrable
 
@@ -524,7 +524,7 @@ theorem AeCover.integral_eq_of_tendsto_of_nonneg_ae [l.ne_bot] [l.IsCountablyGen
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
     (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
     (∫ x, f x ∂μ) = I :=
-  have hfi' : Integrable f μ := hφ.integrableOfIntegralTendstoOfNonnegAe I hfi hnng htendsto
+  have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto
   hφ.integral_eq_of_tendsto I hfi' htendsto
 #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AeCover.integral_eq_of_tendsto_of_nonneg_ae
 
@@ -535,7 +535,7 @@ section IntegrableOfIntervalIntegral
 variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l]
   [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E}
 
-theorem integrableOfIntervalIntegralNormBounded (I : ℝ)
+theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
     (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ :=
   by
@@ -544,21 +544,21 @@ theorem integrableOfIntervalIntegralNormBounded (I : ℝ)
   filter_upwards [ha.eventually (eventually_le_at_bot 0),
     hb.eventually (eventually_ge_at_top 0)]with i hai hbi ht
   rwa [← intervalIntegral.integral_of_le (hai.trans hbi)]
-#align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrableOfIntervalIntegralNormBounded
+#align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded
 
 /-- If `f` is integrable on intervals `Ioc (a i) (b i)`,
 where `a i` tends to -∞ and `b i` tends to ∞, and
 `∫ x in a i .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
 then `f` is integrable on the interval (-∞, ∞) -/
-theorem integrableOfIntervalIntegralNormTendsto (I : ℝ)
+theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
     (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a i..b i, ‖f x‖ ∂μ) l (𝓝 I)) :
     Integrable f μ :=
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
-  integrableOfIntervalIntegralNormBounded I' hfi ha hb hI'
-#align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrableOfIntervalIntegralNormTendsto
+  integrable_of_intervalIntegral_norm_bounded I' hfi ha hb hI'
+#align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrable_of_intervalIntegral_norm_tendsto
 
-theorem integrableOnIicOfIntervalIntegralNormBounded (I b : ℝ)
+theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
     (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ :=
   by
@@ -572,20 +572,20 @@ theorem integrableOnIicOfIntervalIntegralNormBounded (I b : ℝ)
   filter_upwards [ha.eventually (eventually_le_at_bot b)]with i hai
   rw [intervalIntegral.integral_of_le hai, measure.restrict_restrict (hφ.measurable i)]
   exact id
-#align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOnIicOfIntervalIntegralNormBounded
+#align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
 
 /-- If `f` is integrable on intervals `Ioc (a i) b`,
 where `a i` tends to -∞, and
 `∫ x in a i .. b, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
 then `f` is integrable on the interval (-∞, b) -/
-theorem integrableOnIicOfIntervalIntegralNormTendsto (I b : ℝ)
+theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
     (h : Tendsto (fun i => ∫ x in a i..b, ‖f x‖ ∂μ) l (𝓝 I)) : IntegrableOn f (Iic b) μ :=
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
-  integrableOnIicOfIntervalIntegralNormBounded I' b hfi ha hI'
-#align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOnIicOfIntervalIntegralNormTendsto
+  integrableOn_Iic_of_intervalIntegral_norm_bounded I' b hfi ha hI'
+#align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto
 
-theorem integrableOnIoiOfIntervalIntegralNormBounded (I a : ℝ)
+theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
     (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ :=
   by
@@ -599,26 +599,26 @@ theorem integrableOnIoiOfIntervalIntegralNormBounded (I a : ℝ)
   filter_upwards [hb.eventually (eventually_ge_at_top a)]with i hbi
   rw [intervalIntegral.integral_of_le hbi, measure.restrict_restrict (hφ.measurable i), inter_comm]
   exact id
-#align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOnIoiOfIntervalIntegralNormBounded
+#align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
 
 /-- If `f` is integrable on intervals `Ioc a (b i)`,
 where `b i` tends to ∞, and
 `∫ x in a .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
 then `f` is integrable on the interval (a, ∞) -/
-theorem integrableOnIoiOfIntervalIntegralNormTendsto (I a : ℝ)
+theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
     (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
     (h : Tendsto (fun i => ∫ x in a..b i, ‖f x‖ ∂μ) l (𝓝 <| I)) : IntegrableOn f (Ioi a) μ :=
   let ⟨I', hI'⟩ := h.isBoundedUnder_le
-  integrableOnIoiOfIntervalIntegralNormBounded I' a hfi hb hI'
-#align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOnIoiOfIntervalIntegralNormTendsto
+  integrableOn_Ioi_of_intervalIntegral_norm_bounded I' a hfi hb hI'
+#align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
 
-theorem integrableOnIocOfIntervalIntegralNormBounded {I a₀ b₀ : ℝ}
+theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
     (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) :
     IntegrableOn f (Ioc a₀ b₀) :=
   by
   refine'
-    (ae_cover_Ioc_of_Ioc ha hb).integrableOfIntegralNormBounded I
+    (ae_cover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I
       (fun i => (hfi i).restrict measurableSet_Ioc) (eventually.mono h _)
   intro i hi; simp only [measure.restrict_restrict measurableSet_Ioc]
   refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi
@@ -627,19 +627,19 @@ theorem integrableOnIocOfIntervalIntegralNormBounded {I a₀ b₀ : ℝ}
   · apply ae_of_all
     intro c hc
     exact hc.1
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOnIocOfIntervalIntegralNormBounded
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded
 
-theorem integrableOnIocOfIntervalIntegralNormBoundedLeft {I a₀ b : ℝ}
+theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
     (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) :=
-  integrableOnIocOfIntervalIntegralNormBounded hfi ha tendsto_const_nhds h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOnIocOfIntervalIntegralNormBoundedLeft
+  integrableOn_Ioc_of_interval_integral_norm_bounded hfi ha tendsto_const_nhds h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left
 
-theorem integrableOnIocOfIntervalIntegralNormBoundedRight {I a b₀ : ℝ}
+theorem integrableOn_Ioc_of_interval_integral_norm_bounded_right {I a b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
     (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) :=
-  integrableOnIocOfIntervalIntegralNormBounded hfi tendsto_const_nhds hb h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOnIocOfIntervalIntegralNormBoundedRight
+  integrableOn_Ioc_of_interval_integral_norm_bounded hfi tendsto_const_nhds hb h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
 
 end IntegrableOfIntervalIntegral
 
@@ -730,7 +730,7 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDer
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `(a, +∞)` and continuity on `[a, +∞)`. -/
-theorem integrableOnIoiDerivOfNonneg (hcont : ContinuousOn g (Ici a))
+theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
     (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
   by
@@ -738,7 +738,7 @@ theorem integrableOnIoiDerivOfNonneg (hcont : ContinuousOn g (Ici a))
   swap
   ·
     exact
-      intervalIntegral.integrableOnDerivOfNonneg (hcont.mono Icc_subset_Ici_self)
+      intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
         (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
   apply tendsto.congr' _ (hg.sub_const _)
   filter_upwards [Ioi_mem_at_top a]with x hx
@@ -751,7 +751,7 @@ theorem integrableOnIoiDerivOfNonneg (hcont : ContinuousOn g (Ici a))
           fun y hy => hderiv y hy.1
       rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
       exact
-        intervalIntegral.integrableOnDerivOfNonneg (hcont.mono Icc_subset_Ici_self)
+        intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
           (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
     _ = ∫ y in a..id x, ‖g' y‖ :=
       by
@@ -761,18 +761,18 @@ theorem integrableOnIoiDerivOfNonneg (hcont : ContinuousOn g (Ici a))
       rw [abs_of_nonneg]
       exact g'pos _ hy.1
     
-#align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOnIoiDerivOfNonneg
+#align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
 
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
-theorem integrableOnIoiDerivOfNonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
   by
   apply integrable_on_Ioi_deriv_of_nonneg _ (fun x hx => hderiv x (le_of_lt hx)) g'pos hg
   intro x hx
   exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
-#align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOnIoiDerivOfNonneg'
+#align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOn_Ioi_deriv_of_nonneg'
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
@@ -782,7 +782,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
     (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
-    (integrableOnIoiDerivOfNonneg hcont hderiv g'pos hg) hg
+    (integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
@@ -790,13 +790,14 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
-  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOnIoiDerivOfNonneg' hderiv g'pos hg) hg
+  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonneg' hderiv g'pos hg)
+    hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
 
 /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `(a, +∞)` and continuity on `[a, +∞)`. -/
-theorem integrableOnIoiDerivOfNonpos (hcont : ContinuousOn g (Ici a))
+theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
     (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
   by
@@ -804,18 +805,18 @@ theorem integrableOnIoiDerivOfNonpos (hcont : ContinuousOn g (Ici a))
   exact
     integrable_on_Ioi_deriv_of_nonneg hcont.neg (fun x hx => (hderiv x hx).neg)
       (fun x hx => neg_nonneg_of_nonpos (g'neg x hx)) hg.neg
-#align measure_theory.integrable_on_Ioi_deriv_of_nonpos MeasureTheory.integrableOnIoiDerivOfNonpos
+#align measure_theory.integrable_on_Ioi_deriv_of_nonpos MeasureTheory.integrableOn_Ioi_deriv_of_nonpos
 
 /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
-theorem integrableOnIoiDerivOfNonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
   by
   apply integrable_on_Ioi_deriv_of_nonpos _ (fun x hx => hderiv x (le_of_lt hx)) g'neg hg
   intro x hx
   exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
-#align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOnIoiDerivOfNonpos'
+#align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
@@ -825,7 +826,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
     (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
-    (integrableOnIoiDerivOfNonpos hcont hderiv g'neg hg) hg
+    (integrableOn_Ioi_deriv_of_nonpos hcont hderiv g'neg hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
@@ -833,7 +834,8 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
-  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOnIoiDerivOfNonpos' hderiv g'neg hg) hg
+  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg)
+    hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
 
 end IoiFTC

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b8ba04e2f326f3f7cf24ad129beda58531ada61

@@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit d4817f8867c368d6c5571f7379b3888aaec1d95a
+! leanprover-community/mathlib commit 011cafb4a5bc695875d186e245d6b3df03bf6c40
 ! 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.MeasureTheory.Integral.FundThmCalculus
 import Mathbin.Order.Filter.AtTopBot
 import Mathbin.MeasureTheory.Function.Jacobian
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e281deff072232a3c5b3e90034bd65dde396312

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit d4817f8867c368d6c5571f7379b3888aaec1d95a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -55,7 +55,15 @@ as an `ae_cover` w.r.t. `μ.restrict (Iic b)`, instead of using `(λ x, Ioc x b)
   then `∫ x in φ n, f x ∂μ` tends to `∫ x, f x ∂μ` as `n` tends to `+∞`.
 
 We then specialize these lemmas to various use cases involving intervals, which are frequent
-in analysis.
+in analysis. In particular,
+- `measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto` is a version of FTC-2 on the interval
+  `(a, +∞)`, giving the formula `∫ x in (a, +∞), g' x = l - g a` if `g'` is integrable and
+  `g` tends to `l` at `+∞`.
+- `measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg` gives the same result assuming that
+  `g'` is nonnegative instead of integrable. Its automatic integrability in this context is proved
+  in `measure_theory.integrable_on_Ioi_deriv_of_nonneg`.
+- `measure_theory.integral_comp_smul_deriv_Ioi` is a version of the change of variables formula
+  on semi-infinite intervals.
 -/
 
 
@@ -677,14 +685,166 @@ theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (I
 
 end IntegralOfIntervalIntegral
 
+open Real
+
+open Interval
+
+section IoiFTC
+
+variable {E : Type _} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
+  [NormedSpace ℝ E] [CompleteSpace E]
+
+/-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
+When a function has a limit at infinity `m`, and its derivative is integrable, then the
+integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
+on `(a, +∞)` and continuity on `[a, +∞)`.-/
+theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
+    (hf : Tendsto f atTop (𝓝 m)) : (∫ x in Ioi a, f' x) = m - f a :=
+  by
+  refine' tendsto_nhds_unique (interval_integral_tendsto_integral_Ioi a f'int tendsto_id) _
+  apply tendsto.congr' _ (hf.sub_const _)
+  filter_upwards [Ioi_mem_at_top a]with x hx
+  have h'x : a ≤ id x := le_of_lt hx
+  symm
+  apply
+    intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self)
+      fun y hy => hderiv y hy.1
+  rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+  exact f'int.mono (fun y hy => hy.1) le_rfl
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
+
+/-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
+When a function has a limit at infinity `m`, and its derivative is integrable, then the
+integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
+on `[a, +∞)`. -/
+theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
+    (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
+    (∫ x in Ioi a, f' x) = m - f a :=
+  by
+  apply integral_Ioi_of_has_deriv_at_of_tendsto _ (fun x hx => hderiv x (le_of_lt hx)) f'int hf
+  intro x hx
+  exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto' MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto'
+
+/-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
+is automatically integrable on `(a, +∞)`. Version assuming differentiability
+on `(a, +∞)` and continuity on `[a, +∞)`. -/
+theorem integrableOnIoiDerivOfNonneg (hcont : ContinuousOn g (Ici a))
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
+    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
+  by
+  apply integrable_on_Ioi_of_interval_integral_norm_tendsto (l - g a) a (fun x => _) tendsto_id;
+  swap
+  ·
+    exact
+      intervalIntegral.integrableOnDerivOfNonneg (hcont.mono Icc_subset_Ici_self)
+        (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
+  apply tendsto.congr' _ (hg.sub_const _)
+  filter_upwards [Ioi_mem_at_top a]with x hx
+  have h'x : a ≤ id x := le_of_lt hx
+  calc
+    g x - g a = ∫ y in a..id x, g' y := by
+      symm
+      apply
+        intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self)
+          fun y hy => hderiv y hy.1
+      rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+      exact
+        intervalIntegral.integrableOnDerivOfNonneg (hcont.mono Icc_subset_Ici_self)
+          (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
+    _ = ∫ y in a..id x, ‖g' y‖ :=
+      by
+      simp_rw [intervalIntegral.integral_of_le h'x]
+      refine' set_integral_congr measurableSet_Ioc fun y hy => _
+      dsimp
+      rw [abs_of_nonneg]
+      exact g'pos _ hy.1
+    
+#align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOnIoiDerivOfNonneg
+
+/-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
+is automatically integrable on `(a, +∞)`. Version assuming differentiability
+on `[a, +∞)`. -/
+theorem integrableOnIoiDerivOfNonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
+  by
+  apply integrable_on_Ioi_deriv_of_nonneg _ (fun x hx => hderiv x (le_of_lt hx)) g'pos hg
+  intro x hx
+  exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
+#align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOnIoiDerivOfNonneg'
+
+/-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
+integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
+`integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
+continuity on `[a, +∞)`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
+    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+  integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
+    (integrableOnIoiDerivOfNonneg hcont hderiv g'pos hg) hg
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
+
+/-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
+integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
+`integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOnIoiDerivOfNonneg' hderiv g'pos hg) hg
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
+
+/-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
+is automatically integrable on `(a, +∞)`. Version assuming differentiability
+on `(a, +∞)` and continuity on `[a, +∞)`. -/
+theorem integrableOnIoiDerivOfNonpos (hcont : ContinuousOn g (Ici a))
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
+    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
+  by
+  apply integrable_neg_iff.1
+  exact
+    integrable_on_Ioi_deriv_of_nonneg hcont.neg (fun x hx => (hderiv x hx).neg)
+      (fun x hx => neg_nonneg_of_nonpos (g'neg x hx)) hg.neg
+#align measure_theory.integrable_on_Ioi_deriv_of_nonpos MeasureTheory.integrableOnIoiDerivOfNonpos
+
+/-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
+is automatically integrable on `(a, +∞)`. Version assuming differentiability
+on `[a, +∞)`. -/
+theorem integrableOnIoiDerivOfNonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) :=
+  by
+  apply integrable_on_Ioi_deriv_of_nonpos _ (fun x hx => hderiv x (le_of_lt hx)) g'neg hg
+  intro x hx
+  exact (hderiv x hx).ContinuousAt.ContinuousWithinAt
+#align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOnIoiDerivOfNonpos'
+
+/-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
+integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
+`integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
+continuity on `[a, +∞)`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
+    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+  integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
+    (integrableOnIoiDerivOfNonpos hcont hderiv g'neg hg) hg
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
+
+/-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
+integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
+`integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
+    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+  integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOnIoiDerivOfNonpos' hderiv g'neg hg) hg
+#align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
+
+end IoiFTC
+
 section IoiChangeVariables
 
 open Real
 
 open Interval
 
-variable {E : Type _} {μ : Measure ℝ} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [CompleteSpace E]
+variable {E : Type _} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
 /-- Change-of-variables formula for `Ioi` integrals of vector-valued functions, proved by taking
 limits from the result for finite intervals. -/

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

@@ -61,7 +61,7 @@ in analysis.
 
 open MeasureTheory Filter Set TopologicalSpace
 
-open Ennreal NNReal Topology
+open ENNReal NNReal Topology
 
 namespace MeasureTheory
 
@@ -410,20 +410,20 @@ variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter 
 
 theorem AeCover.integrableOfLintegralNnnormBounded [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
-    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ Ennreal.ofReal I) : Integrable f μ :=
+    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ :=
   by
-  refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt Ennreal.ofReal_lt_top⟩
+  refine' ⟨hfm, (le_of_tendsto _ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
   exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AeCover.integrableOfLintegralNnnormBounded
 
 theorem AeCover.integrableOfLintegralNnnormTendsto [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ) (hfm : AeStronglyMeasurable f μ)
-    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| Ennreal.ofReal I)) :
+    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
     Integrable f μ :=
   by
   refine' hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm _
   refine' htendsto.eventually (ge_mem_nhds _)
-  refine' (Ennreal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _
+  refine' (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _
   exact lt_max_of_lt_right (lt_add_one I)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AeCover.integrableOfLintegralNnnormTendsto
 
@@ -431,14 +431,14 @@ theorem AeCover.integrableOfLintegralNnnormBounded' [l.ne_bot] [l.IsCountablyGen
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
     (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
   hφ.integrableOfLintegralNnnormBounded I hfm
-    (by simpa only [Ennreal.ofReal_coe_nNReal] using hbounded)
+    (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AeCover.integrableOfLintegralNnnormBounded'
 
 theorem AeCover.integrableOfLintegralNnnormTendsto' [l.ne_bot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AeCover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AeStronglyMeasurable f μ)
     (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
   hφ.integrableOfLintegralNnnormTendsto I hfm
-    (by simpa only [Ennreal.ofReal_coe_nNReal] using htendsto)
+    (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
 #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AeCover.integrableOfLintegralNnnormTendsto'
 
 theorem AeCover.integrableOfIntegralNormBounded [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}
@@ -451,13 +451,13 @@ theorem AeCover.integrableOfIntegralNormBounded [l.ne_bot] [l.IsCountablyGenerat
   conv at hbounded in integral _ _ =>
     rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x))
         hfm.norm.restrict]
-  conv at hbounded in Ennreal.ofReal _ =>
+  conv at hbounded in ENNReal.ofReal _ =>
     dsimp
     rw [← coe_nnnorm]
-    rw [Ennreal.ofReal_coe_nNReal]
+    rw [ENNReal.ofReal_coe_nnreal]
   refine' hbounded.mono fun i hi => _
-  rw [← Ennreal.ofReal_toReal (ne_top_of_lt (hfi i).2)]
-  apply Ennreal.ofReal_le_ofReal hi
+  rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)]
+  apply ENNReal.ofReal_le_ofReal hi
 #align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AeCover.integrableOfIntegralNormBounded
 
 theorem AeCover.integrableOfIntegralNormTendsto [l.ne_bot] [l.IsCountablyGenerated] {φ : ι → Set α}

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

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -816,7 +816,7 @@ theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
     (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by
   have := fun x (_ : x ∈ Ioi b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
   rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub]
-  refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
+  · refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
   rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
   exact h2f.filter_mono _root_.atTop_le_cocompact |>.tendsto
 
@@ -1019,7 +1019,7 @@ theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
     (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Iic b, deriv f x = f b := by
   have := fun x (_ : x ∈ Iio b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
   rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
-  refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
+  · refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
   rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
   exact h2f.filter_mono _root_.atBot_le_cocompact |>.tendsto
 

@@ -761,8 +761,7 @@ theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
     apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A
     intro s hs
     rcases mem_atTop_sets.1 hs with ⟨b, hb⟩
-    apply le_antisymm (le_top)
-    rw [← volume_Ici (a := b)]
+    rw [← top_le_iff, ← volume_Ici (a := b)]
     exact measure_mono hb
   rwa [B, ← Embedding.tendsto_nhds_iff] at A
   exact (Completion.uniformEmbedding_coe E).embedding
@@ -938,8 +937,8 @@ theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic [CompleteSpace E]
   suffices ∃ a, Tendsto f atBot (𝓝 a) from tendsto_nhds_limUnder this
   let g := f ∘ (fun x ↦ -x)
   have hdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x := by
-    intro x (hx : -a < x)
-    have : -x ∈ Iic a := by simp; linarith
+    intro x hx
+    have : -x ∈ Iic a := by simp only [mem_Iic, mem_Ioi, neg_le] at *; exact hx.le
     simpa using HasDerivAt.scomp x (hderiv (-x) this) (hasDerivAt_neg' x)
   have L : Tendsto g atTop (𝓝 (limUnder atTop g)) := by
     apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi hdg

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

@@ -622,7 +622,7 @@ theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ}
   refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I
     (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi ↦ ?_)
   rw [Measure.restrict_restrict measurableSet_Ioc]
-  refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi <;> apply ae_of_all
+  refine' le_trans (setIntegral_mono_set (hfi i).norm _ _) hi <;> apply ae_of_all
   · simp only [Pi.zero_apply, norm_nonneg, forall_const]
   · intro c hc; exact hc.1
 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
@@ -708,7 +708,7 @@ theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
   have A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε := by
     have L : Tendsto (fun (n : ℕ) ↦ ∫ x in Ici (n : ℝ), ‖f' x‖) atTop
         (𝓝 (∫ x in ⋂ (n : ℕ), Ici (n : ℝ), ‖f' x‖)) := by
-      apply tendsto_set_integral_of_antitone (fun n ↦ measurableSet_Ici)
+      apply tendsto_setIntegral_of_antitone (fun n ↦ measurableSet_Ici)
       · intro m n hmn
         exact Ici_subset_Ici.2 (Nat.cast_le.mpr hmn)
       · rcases exists_nat_gt a with ⟨n, hn⟩
@@ -735,7 +735,7 @@ theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
         exact f'int.mono_set (Ioc_subset_Ioi_self.trans (Ioi_subset_Ioi h'N.le))
   _ ≤ ∫ t in Ioc ↑N x, ‖f' t‖ := norm_integral_le_integral_norm fun a ↦ f' a
   _ ≤ ∫ t in Ici ↑N, ‖f' t‖ := by
-      apply set_integral_mono_set
+      apply setIntegral_mono_set
       · apply IntegrableOn.mono_set f'int.norm (Ici_subset_Ioi.2 h'N)
       · filter_upwards with x using norm_nonneg _
       · have : Ioc (↑N) x ⊆ Ici ↑N := Ioc_subset_Ioi_self.trans Ioi_subset_Ici_self
@@ -848,7 +848,7 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a
         (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
     _ = ∫ y in a..id x, ‖g' y‖ := by
       simp_rw [intervalIntegral.integral_of_le h'x]
-      refine' set_integral_congr measurableSet_Ioc fun y hy => _
+      refine' setIntegral_congr measurableSet_Ioc fun y hy => _
       dsimp
       rw [abs_of_nonneg]
       exact g'pos _ hy.1
@@ -1133,7 +1133,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g
   rw [a3] at this; rw [this]
-  refine' set_integral_congr measurableSet_Ioi _
+  refine' setIntegral_congr measurableSet_Ioi _
   intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg (le_of_lt hx) _)]
 #align measure_theory.integral_comp_rpow_Ioi MeasureTheory.integral_comp_rpow_Ioi

@@ -615,7 +615,7 @@ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
   integrableOn_Ioi_of_intervalIntegral_norm_bounded I' a hfi hb hI'
 #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
 
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
     (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) :
     IntegrableOn f (Ioc a₀ b₀) := by
@@ -625,19 +625,26 @@ theorem integrableOn_Ioc_of_interval_integral_norm_bounded {I a₀ b₀ : ℝ}
   refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi <;> apply ae_of_all
   · simp only [Pi.zero_apply, norm_nonneg, forall_const]
   · intro c hc; exact hc.1
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
 
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded_left {I a₀ b : ℝ}
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
     (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) :=
-  integrableOn_Ioc_of_interval_integral_norm_bounded hfi ha tendsto_const_nhds h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi ha tendsto_const_nhds h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left
 
-theorem integrableOn_Ioc_of_interval_integral_norm_bounded_right {I a b₀ : ℝ}
+theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ}
     (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
     (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) :=
-  integrableOn_Ioc_of_interval_integral_norm_bounded hfi tendsto_const_nhds hb h
-#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi tendsto_const_nhds hb h
+#align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right
+
+@[deprecated] alias integrableOn_Ioc_of_interval_integral_norm_bounded :=
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded -- 2024-04-06
+@[deprecated] alias integrableOn_Ioc_of_interval_integral_norm_bounded_left :=
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded_left -- 2024-04-06
+@[deprecated] alias integrableOn_Ioc_of_interval_integral_norm_bounded_right :=
+  integrableOn_Ioc_of_intervalIntegral_norm_bounded_right -- 2024-04-06
 
 end IntegrableOfIntervalIntegral
 

We already have that ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x if u * v tends to a' and b' at minus infinity and infinity. Assuming morevoer that u * v is integrable, we show that it tends to 0 at minus infinity and infinity, and therefore that ∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x. We also give versions with a general bilinear form instead of multiplication.

@@ -9,6 +9,7 @@ import Mathlib.MeasureTheory.Integral.FundThmCalculus
 import Mathlib.Order.Filter.AtTopBot
 import Mathlib.MeasureTheory.Function.Jacobian
 import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
+import Mathlib.MeasureTheory.Measure.Haar.Unique
 
 #align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
 
@@ -68,6 +69,13 @@ in analysis. In particular,
   in `MeasureTheory.integrableOn_Ioi_deriv_of_nonneg`.
 - `MeasureTheory.integral_comp_smul_deriv_Ioi` is a version of the change of variables formula
   on semi-infinite intervals.
+- `MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi` shows that a function whose
+  derivative is integrable on `(a, +∞)` has a limit at `+∞`.
+- `MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi` shows that an integrable function
+  whose derivative is integrable on `(a, +∞)` tends to `0` at `+∞`.
+
+Versions of these results are also given on the intervals `(-∞, a]` and `(-∞, +∞)`, as well as
+the corresponding versions of integration by parts.
 -/
 
 open MeasureTheory Filter Set TopologicalSpace
@@ -680,12 +688,87 @@ open scoped Interval
 section IoiFTC
 
 variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
-  [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedSpace ℝ E]
+
+/-- If the derivative of a function defined on the real line is integrable close to `+∞`, then
+the function has a limit at `+∞`. -/
+theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) :
+    Tendsto f atTop (𝓝 (limUnder atTop f)) := by
+  suffices ∃ a, Tendsto f atTop (𝓝 a) from tendsto_nhds_limUnder this
+  suffices CauchySeq f from cauchySeq_tendsto_of_complete this
+  apply Metric.cauchySeq_iff'.2 (fun ε εpos ↦ ?_)
+  have A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε := by
+    have L : Tendsto (fun (n : ℕ) ↦ ∫ x in Ici (n : ℝ), ‖f' x‖) atTop
+        (𝓝 (∫ x in ⋂ (n : ℕ), Ici (n : ℝ), ‖f' x‖)) := by
+      apply tendsto_set_integral_of_antitone (fun n ↦ measurableSet_Ici)
+      · intro m n hmn
+        exact Ici_subset_Ici.2 (Nat.cast_le.mpr hmn)
+      · rcases exists_nat_gt a with ⟨n, hn⟩
+        exact ⟨n, IntegrableOn.mono_set f'int.norm (Ici_subset_Ioi.2 hn)⟩
+    have B : ⋂ (n : ℕ), Ici (n : ℝ) = ∅ := by
+      apply eq_empty_of_forall_not_mem (fun x ↦ ?_)
+      simpa only [mem_iInter, mem_Ici, not_forall, not_le] using exists_nat_gt x
+    simp only [B, Measure.restrict_empty, integral_zero_measure] at L
+    exact (tendsto_order.1 L).2 _ εpos
+  have B : ∀ᶠ (n : ℕ) in atTop, a < n := by
+    rcases exists_nat_gt a with ⟨n, hn⟩
+    filter_upwards [Ioi_mem_atTop n] with m (hm : n < m) using hn.trans (Nat.cast_lt.mpr hm)
+  rcases (A.and B).exists with ⟨N, hN, h'N⟩
+  refine ⟨N, fun x hx ↦ ?_⟩
+  calc
+  dist (f x) (f ↑N)
+    = ‖f x - f N‖ := dist_eq_norm _ _
+  _ = ‖∫ t in Ioc ↑N x, f' t‖ := by
+      rw [← intervalIntegral.integral_of_le hx, intervalIntegral.integral_eq_sub_of_hasDerivAt]
+      · intro y hy
+        simp only [hx, uIcc_of_le, mem_Icc] at hy
+        exact hderiv _ (h'N.trans_le hy.1)
+      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hx]
+        exact f'int.mono_set (Ioc_subset_Ioi_self.trans (Ioi_subset_Ioi h'N.le))
+  _ ≤ ∫ t in Ioc ↑N x, ‖f' t‖ := norm_integral_le_integral_norm fun a ↦ f' a
+  _ ≤ ∫ t in Ici ↑N, ‖f' t‖ := by
+      apply set_integral_mono_set
+      · apply IntegrableOn.mono_set f'int.norm (Ici_subset_Ioi.2 h'N)
+      · filter_upwards with x using norm_nonneg _
+      · have : Ioc (↑N) x ⊆ Ici ↑N := Ioc_subset_Ioi_self.trans Ioi_subset_Ici_self
+        exact this.eventuallyLE
+  _ < ε := hN
+
+open UniformSpace in
+/-- If a function and its derivative are integrable on `(a, +∞)`, then the function tends to zero
+at `+∞`. -/
+theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
+    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
+    (f'int : IntegrableOn f' (Ioi a)) (fint : IntegrableOn f (Ioi a)) :
+    Tendsto f atTop (𝓝 0) := by
+  let F : E →L[ℝ] Completion E := Completion.toComplL
+  have Fderiv : ∀ x ∈ Ioi a, HasDerivAt (F ∘ f) (F (f' x)) x :=
+    fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx)
+  have Fint : IntegrableOn (F ∘ f) (Ioi a) := by apply F.integrable_comp fint
+  have F'int : IntegrableOn (F ∘ f') (Ioi a) := by apply F.integrable_comp f'int
+  have A : Tendsto (F ∘ f) atTop (𝓝 (limUnder atTop (F ∘ f))) := by
+    apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi Fderiv F'int
+  have B : limUnder atTop (F ∘ f) = F 0 := by
+    have : IntegrableAtFilter (F ∘ f) atTop := by exact ⟨Ioi a, Ioi_mem_atTop _, Fint⟩
+    apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A
+    intro s hs
+    rcases mem_atTop_sets.1 hs with ⟨b, hb⟩
+    apply le_antisymm (le_top)
+    rw [← volume_Ici (a := b)]
+    exact measure_mono hb
+  rwa [B, ← Embedding.tendsto_nhds_iff] at A
+  exact (Completion.uniformEmbedding_coe E).embedding
+
+variable [CompleteSpace E]
 
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
 When a function has a limit at infinity `m`, and its derivative is integrable, then the
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
-on `(a, +∞)` and continuity at `a⁺`.-/
+on `(a, +∞)` and continuity at `a⁺`.
+
+Note that such a function always has a limit at infinity,
+see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
     (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a := by
@@ -709,7 +792,10 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
 When a function has a limit at infinity `m`, and its derivative is integrable, then the
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
-on `[a, +∞)`. -/
+on `[a, +∞)`.
+
+Note that such a function always has a limit at infinity,
+see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
     (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
     ∫ x in Ioi a, f' x = m - f a := by
@@ -835,11 +921,61 @@ end IoiFTC
 section IicFTC
 
 variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
-  [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedSpace ℝ E]
+
+/-- If the derivative of a function defined on the real line is integrable close to `-∞`, then
+the function has a limit at `-∞`. -/
+theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic [CompleteSpace E]
+    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) :
+    Tendsto f atBot (𝓝 (limUnder atBot f)) := by
+  suffices ∃ a, Tendsto f atBot (𝓝 a) from tendsto_nhds_limUnder this
+  let g := f ∘ (fun x ↦ -x)
+  have hdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x := by
+    intro x (hx : -a < x)
+    have : -x ∈ Iic a := by simp; linarith
+    simpa using HasDerivAt.scomp x (hderiv (-x) this) (hasDerivAt_neg' x)
+  have L : Tendsto g atTop (𝓝 (limUnder atTop g)) := by
+    apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi hdg
+    exact ((MeasurePreserving.integrableOn_comp_preimage (Measure.measurePreserving_neg _)
+      (Homeomorph.neg ℝ).measurableEmbedding).2 f'int.neg).mono_set (by simp)
+  refine ⟨limUnder atTop g, ?_⟩
+  have : Tendsto (fun x ↦ g (-x)) atBot (𝓝 (limUnder atTop g)) := L.comp tendsto_neg_atBot_atTop
+  simpa [g] using this
+
+open UniformSpace in
+/-- If a function and its derivative are integrable on `(-∞, a]`, then the function tends to zero
+at `-∞`. -/
+theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
+    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x)
+    (f'int : IntegrableOn f' (Iic a)) (fint : IntegrableOn f (Iic a)) :
+    Tendsto f atBot (𝓝 0) := by
+  let F : E →L[ℝ] Completion E := Completion.toComplL
+  have Fderiv : ∀ x ∈ Iic a, HasDerivAt (F ∘ f) (F (f' x)) x :=
+    fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx)
+  have Fint : IntegrableOn (F ∘ f) (Iic a) := by apply F.integrable_comp fint
+  have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int
+  have A : Tendsto (F ∘ f) atBot (𝓝 (limUnder atBot (F ∘ f))) := by
+    apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int
+  have B : limUnder atBot (F ∘ f) = F 0 := by
+    have : IntegrableAtFilter (F ∘ f) atBot := by exact ⟨Iic a, Iic_mem_atBot _, Fint⟩
+    apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A
+    intro s hs
+    rcases mem_atBot_sets.1 hs with ⟨b, hb⟩
+    apply le_antisymm (le_top)
+    rw [← volume_Iic (a := b)]
+    exact measure_mono hb
+  rwa [B, ← Embedding.tendsto_nhds_iff] at A
+  exact (Completion.uniformEmbedding_coe E).embedding
+
+variable [CompleteSpace E]
+
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
 When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
 integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
-on `(-∞, a)` and continuity at `a⁻`. -/
+on `(-∞, a)` and continuity at `a⁻`.
+
+Note that such a function always has a limit at minus infinity,
+see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/
 theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
     (hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
     (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by
@@ -860,7 +996,10 @@ theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
 When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
 integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
-on `(-∞, a]`. -/
+on `(-∞, a]`.
+
+Note that such a function always has a limit at minus infinity,
+see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/
 theorem integral_Iic_of_hasDerivAt_of_tendsto'
     (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
     (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by
@@ -883,9 +1022,16 @@ end IicFTC
 section UnivFTC
 
 variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m n : E} [NormedAddCommGroup E]
-  [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedSpace ℝ E]
+
+/-- **Fundamental theorem of calculus-2**, on the whole real line
+When a function has a limit `m` at `-∞` and `n` at `+∞`, and its derivative is integrable, then the
+integral of the derivative is `n - m`.
 
-theorem integral_of_hasDerivAt_of_tendsto
+Note that such a function always has a limit at `-∞` and `+∞`,
+see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic` and
+`tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
+theorem integral_of_hasDerivAt_of_tendsto [CompleteSpace E]
     (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f')
     (hbot : Tendsto f atBot (𝓝 m)) (htop : Tendsto f atTop (𝓝 n)) : ∫ x, f' x = n - m := by
   rw [← integral_univ, ← Set.Iic_union_Ioi (a := 0),
@@ -894,6 +1040,21 @@ theorem integral_of_hasDerivAt_of_tendsto
     integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn htop]
   abel
 
+/-- If a function and its derivative are integrable on the real line, then the integral of the
+derivative is zero. -/
+theorem integral_eq_zero_of_hasDerivAt_of_integrable
+    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f') (hf : Integrable f) :
+    ∫ x, f' x = 0 := by
+  by_cases hE : CompleteSpace E; swap
+  · simp [integral, hE]
+  have A : Tendsto f atBot (𝓝 0) :=
+    tendsto_zero_of_hasDerivAt_of_integrableOn_Iic (a := 0) (fun x _hx ↦ hderiv x)
+      hf'.integrableOn hf.integrableOn
+  have B : Tendsto f atTop (𝓝 0) :=
+    tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi (a := 0) (fun x _hx ↦ hderiv x)
+      hf'.integrableOn hf.integrableOn
+  simpa using integral_of_hasDerivAt_of_tendsto hderiv hf' A B
+
 end UnivFTC
 
 section IoiChangeVariables
@@ -1058,13 +1219,63 @@ end IoiIntegrability
 ## Integration by parts
 -/
 
-section IntegrationByParts
+section IntegrationByPartsBilinear
+
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G]
+  {L : E →L[ℝ] F →L[ℝ] G} {u : ℝ → E} {v : ℝ → F} {u' : ℝ → E} {v' : ℝ → F}
+  {m n : G}
 
-variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
+theorem integral_bilinear_hasDerivAt_eq_sub [CompleteSpace G]
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv : Integrable (fun x ↦ L (u x) (v' x) + L (u' x) (v x)))
+    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
+    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
+    ∫ (x : ℝ), L (u x) (v' x) + L (u' x) (v x) = n - m :=
+  integral_of_hasDerivAt_of_tendsto (fun x ↦ L.hasDerivAt_of_bilinear (hu x) (hv x))
+    huv h_bot h_top
+
+/-- **Integration by parts on (-∞, ∞).**
+With respect to a general bilinear form. For the specific case of multiplication, see
+`integral_mul_deriv_eq_deriv_mul`. -/
+theorem integral_bilinear_hasDerivAt_right_eq_sub [CompleteSpace G]
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
+    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
+    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
+    ∫ (x : ℝ), L (u x) (v' x) = n - m - ∫ (x : ℝ), L (u' x) (v x) := by
+  rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
+  exact integral_bilinear_hasDerivAt_eq_sub hu hv (huv'.add hu'v) h_bot h_top
+
+/-- **Integration by parts on (-∞, ∞).**
+With respect to a general bilinear form, assuming moreover that the total function is integrable.
+-/
+theorem integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
+    (huv : Integrable (fun x ↦ L (u x) (v x))) :
+    ∫ (x : ℝ), L (u x) (v' x) = - ∫ (x : ℝ), L (u' x) (v x) := by
+  by_cases hG : CompleteSpace G; swap
+  · simp [integral, hG]
+  have I : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 0) :=
+    tendsto_zero_of_hasDerivAt_of_integrableOn_Iic (a := 0)
+      (fun x _hx ↦ L.hasDerivAt_of_bilinear (hu x) (hv x))
+      (huv'.add hu'v).integrableOn huv.integrableOn
+  have J : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 0) :=
+    tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi (a := 0)
+      (fun x _hx ↦ L.hasDerivAt_of_bilinear (hu x) (hv x))
+      (huv'.add hu'v).integrableOn huv.integrableOn
+  simp [integral_bilinear_hasDerivAt_right_eq_sub hu hv huv' hu'v I J]
+
+end IntegrationByPartsBilinear
+
+section IntegrationByPartsAlgebra
+
+variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A]
   {a b : ℝ} {a' b' : A} {u : ℝ → A} {v : ℝ → A} {u' : ℝ → A} {v' : ℝ → A}
 
 /-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
-theorem integral_deriv_mul_eq_sub
+theorem integral_deriv_mul_eq_sub [CompleteSpace A]
     (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
     (huv : Integrable (u' * v + u * v'))
     (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
@@ -1073,14 +1284,24 @@ theorem integral_deriv_mul_eq_sub
 
 /-- **Integration by parts on (-∞, ∞).**
 For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
-theorem integral_mul_deriv_eq_deriv_mul
+theorem integral_mul_deriv_eq_deriv_mul [CompleteSpace A]
     (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
     (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v))
     (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
-    ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x := by
-  rw [Pi.mul_def] at huv' hu'v
-  rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
-  simpa only [add_comm] using integral_deriv_mul_eq_sub hu hv (hu'v.add huv') h_bot h_top
+    ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x :=
+  integral_bilinear_hasDerivAt_right_eq_sub  (L := ContinuousLinearMap.mul ℝ A)
+    hu hv huv' hu'v h_bot h_top
+
+/-- **Integration by parts on (-∞, ∞).**
+Version assuming that the total function is integrable -/
+theorem integral_mul_deriv_eq_deriv_mul_of_integrable
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v)) (huv : Integrable (u * v)) :
+    ∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x :=
+  integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable (L := ContinuousLinearMap.mul ℝ A)
+    hu hv huv' hu'v huv
+
+variable [CompleteSpace A]
 
 -- TODO: also apply `Tendsto _ (𝓝[>] a) (𝓝 a')` generalization to
 -- `integral_Ioi_of_hasDerivAt_of_tendsto` and `integral_Iic_of_hasDerivAt_of_tendsto`
@@ -1146,6 +1367,6 @@ theorem integral_Iic_mul_deriv_eq_deriv_mul
   rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
   simpa only [add_comm] using integral_Iic_deriv_mul_eq_sub hu hv (hu'v.add huv') h_zero h_infty
 
-end IntegrationByParts
+end IntegrationByPartsAlgebra
 
 end MeasureTheory

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 | |

@@ -983,7 +983,7 @@ theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 <
     ← abs_of_pos (inv_pos.mpr hb), ← Measure.integral_comp_mul_left]
   congr
   ext1 x
-  rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left _ hb.ne']
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left₀ _ hb.ne']
   rfl
 #align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
 
@@ -1042,7 +1042,8 @@ theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (
   rw [← integrable_indicator_iff (measurableSet_Ioi : MeasurableSet <| Ioi <| a * c)]
   convert integrable_comp_mul_left_iff ((Ioi (a * c)).indicator f) ha.ne' using 2
   ext1 x
-  rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c, mul_div_cancel _ ha.ne']
+  rw [← indicator_comp_right, preimage_const_mul_Ioi _ ha, mul_comm a c,
+    mul_div_cancel_right₀ _ ha.ne']
   rfl
 #align measure_theory.integrable_on_Ioi_comp_mul_left_iff MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
 

This is presumably not exhaustive, but covers about a hundred instances.

Style opinions (e.g., why a particular change is great/not a good idea) are very welcome; I'm still forming my own.

@@ -132,14 +132,14 @@ theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
     AECover μ l (fun i ↦ Metric.ball x (r i)) where
   measurableSet _ := Metric.isOpen_ball.measurableSet
   ae_eventually_mem := by
-    apply eventually_of_forall (fun y ↦ ?_)
+    filter_upwards with y
     filter_upwards [hr (Ioi_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
 
 theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
     AECover μ l (fun i ↦ Metric.closedBall x (r i)) where
   measurableSet _ := Metric.isClosed_ball.measurableSet
   ae_eventually_mem := by
-    apply eventually_of_forall (fun y ↦ ?_)
+    filter_upwards with y
     filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
 
 end MetricSpace

More tactics that are not used, found using the linter at #11308.

The PR consists of tactic removals, whitespace changes and replacing a porting note by an explanation.

@@ -973,7 +973,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
 theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
     (∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y := by
   convert integral_comp_rpow_Ioi g hp.ne'
-  funext; congr; rw [abs_of_nonneg hp.le]
+  rw [abs_of_nonneg hp.le]
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
 
 theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :

Per the style guidelines, λ is disallowed in mathlib. This is close to exhaustive; I left some tactic code alone when it seemed to me that tactic could be upstreamed soon.

Notes

• In lines I was modifying anyway, I also converted => to ↦.
• Also contains some mild in-passing indentation fixes in Mathlib/Order/SupClosed.
• Some doc comments still contained Lean 3 syntax λ x, , which I also replaced.

@@ -39,7 +39,7 @@ in φ i, f x ∂μ` as `i` tends to `l`.
 When using this definition with a measure restricted to a set `s`, which happens fairly often, one
 should not try too hard to use a `MeasureTheory.AECover` of subsets of `s`, as it often makes proofs
 more complicated than necessary. See for example the proof of
-`MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(λ x, Ioi x)` as a
+`MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(fun x ↦ oi x)` as a
 `MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(fun x ↦ Ioc x b)`.
 
 ## Main statements

I removed some of the tactics that were not used and are hopefully uncontroversial arising from the linter at #11308.

As the commit messages should convey, the removed tactics are, essentially,

push_cast
norm_cast
congr
norm_num
dsimp
funext
intro
infer_instance

@@ -464,7 +464,6 @@ theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGene
     rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x))
         hfm.norm.restrict]
   conv at hbounded in ENNReal.ofReal _ =>
-    dsimp
     rw [← coe_nnnorm]
     rw [ENNReal.ofReal_coe_nnreal]
   refine' hbounded.mono fun i hi => _

Classifies by adding issue number #10756 to porting notes claiming anything semantically equivalent to:

• "new lemma"
• "added lemma"

@@ -195,21 +195,21 @@ section FiniteIntervals
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B))
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) where
   ae_eventually_mem := (ae_restrict_mem measurableSet_Ioi).mono fun _x hx ↦ ha.eventually <|
     eventually_lt_nhds hx
   measurableSet _ := measurableSet_Ioi
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) :=
   aecover_Ioi_of_Ioi (α := αᵒᵈ) hb
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) :=
   (aecover_Ioi_of_Ioi ha).superset (fun _ ↦ Ioi_subset_Ici_self) fun _ ↦ measurableSet_Ici
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) :=
   aecover_Ioi_of_Ici (α := αᵒᵈ) hb
 

Co-authored-by: L Lllvvuu <git@llllvvuu.dev> Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: L <git@llllvvuu.dev>

@@ -1054,4 +1054,98 @@ theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ}
 
 end IoiIntegrability
 
+/-!
+## Integration by parts
+-/
+
+section IntegrationByParts
+
+variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
+  {a b : ℝ} {a' b' : A} {u : ℝ → A} {v : ℝ → A} {u' : ℝ → A} {v' : ℝ → A}
+
+/-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
+theorem integral_deriv_mul_eq_sub
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv : Integrable (u' * v + u * v'))
+    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
+    ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a' :=
+  integral_of_hasDerivAt_of_tendsto (fun x ↦ (hu x).mul (hv x)) huv h_bot h_top
+
+/-- **Integration by parts on (-∞, ∞).**
+For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
+theorem integral_mul_deriv_eq_deriv_mul
+    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
+    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v))
+    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
+    ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x := by
+  rw [Pi.mul_def] at huv' hu'v
+  rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
+  simpa only [add_comm] using integral_deriv_mul_eq_sub hu hv (hu'v.add huv') h_bot h_top
+
+-- TODO: also apply `Tendsto _ (𝓝[>] a) (𝓝 a')` generalization to
+-- `integral_Ioi_of_hasDerivAt_of_tendsto` and `integral_Iic_of_hasDerivAt_of_tendsto`
+/-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
+theorem integral_Ioi_deriv_mul_eq_sub
+    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
+    (huv : IntegrableOn (u' * v + u * v') (Ioi a))
+    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
+    ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a' := by
+  rw [← Ici_diff_left] at h_zero
+  let f := Function.update (u * v) a a'
+  have hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x := by
+    intro x (hx : a < x)
+    apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
+    filter_upwards [eventually_ne_nhds hx.ne.symm] with y hy
+    exact Function.update_noteq hy a' (u * v)
+  have htendsto : Tendsto f atTop (𝓝 b') := by
+    apply h_infty.congr'
+    filter_upwards [eventually_ne_atTop a] with x hx
+    exact (Function.update_noteq hx a' (u * v)).symm
+  simpa using integral_Ioi_of_hasDerivAt_of_tendsto
+    (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto
+
+/-- **Integration by parts on (a, ∞).**
+For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
+theorem integral_Ioi_mul_deriv_eq_deriv_mul
+    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
+    (huv' : IntegrableOn (u * v') (Ioi a)) (hu'v : IntegrableOn (u' * v) (Ioi a))
+    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
+    ∫ (x : ℝ) in Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Ioi a, u' x * v x := by
+  rw [Pi.mul_def] at huv' hu'v
+  rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
+  simpa only [add_comm] using integral_Ioi_deriv_mul_eq_sub hu hv (hu'v.add huv') h_zero h_infty
+
+/-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
+theorem integral_Iic_deriv_mul_eq_sub
+    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
+    (huv : IntegrableOn (u' * v + u * v') (Iic a))
+    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
+    ∫ (x : ℝ) in Iic a, u' x * v x + u x * v' x = a' - b' := by
+  rw [← Iic_diff_right] at h_zero
+  let f := Function.update (u * v) a a'
+  have hderiv : ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x := by
+    intro x hx
+    apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
+    filter_upwards [Iio_mem_nhds hx] with x (hx : x < a)
+    exact Function.update_noteq (ne_of_lt hx) a' (u * v)
+  have htendsto : Tendsto f atBot (𝓝 b') := by
+    apply h_infty.congr'
+    filter_upwards [Iio_mem_atBot a] with x (hx : x < a)
+    exact (Function.update_noteq (ne_of_lt hx) a' (u * v)).symm
+  simpa using integral_Iic_of_hasDerivAt_of_tendsto
+    (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto
+
+/-- **Integration by parts on (∞, a].**
+For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
+theorem integral_Iic_mul_deriv_eq_deriv_mul
+    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
+    (huv' : IntegrableOn (u * v') (Iic a)) (hu'v : IntegrableOn (u' * v) (Iic a))
+    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
+    ∫ (x : ℝ) in Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Iic a, u' x * v x := by
+  rw [Pi.mul_def] at huv' hu'v
+  rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
+  simpa only [add_comm] using integral_Iic_deriv_mul_eq_sub hu hv (hu'v.add huv') h_zero h_infty
+
+end IntegrationByParts
+
 end MeasureTheory

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -195,21 +195,21 @@ section FiniteIntervals
 variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]
   {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B))
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) where
   ae_eventually_mem := (ae_restrict_mem measurableSet_Ioi).mono fun _x hx ↦ ha.eventually <|
     eventually_lt_nhds hx
   measurableSet _ := measurableSet_Ioi
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) :=
   aecover_Ioi_of_Ioi (α := αᵒᵈ) hb
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) :=
   (aecover_Ioi_of_Ioi ha).superset (fun _ ↦ Ioi_subset_Ici_self) fun _ ↦ measurableSet_Ici
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) :=
   aecover_Ioi_of_Ici (α := αᵒᵈ) hb
 
@@ -358,7 +358,7 @@ theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AE
     fun _ _ ↦ (hφ.2 _)
 #align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AECover.biUnion_Iic_aecover
 
--- porting note: generalized from `[SemilatticeSup ι] [Nonempty ι]` to `[Preorder ι]`
+-- Porting note: generalized from `[SemilatticeSup ι] [Nonempty ι]` to `[Preorder ι]`
 theorem AECover.biInter_Ici_aecover [Preorder ι] {φ : ι → Set α}
     (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (_h : k ∈ Ici n), φ k where
   ae_eventually_mem := hφ.ae_eventually_mem.mono fun x h ↦ by

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -978,10 +978,10 @@ theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
 #align measure_theory.integral_comp_rpow_Ioi_of_pos MeasureTheory.integral_comp_rpow_Ioi_of_pos
 
 theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
-    (∫ x in Ioi a, g (b * x)) = |b⁻¹| • ∫ x in Ioi (b * a), g x := by
+    (∫ x in Ioi a, g (b * x)) = b⁻¹ • ∫ x in Ioi (b * a), g x := by
   have : ∀ c : ℝ, MeasurableSet (Ioi c) := fun c => measurableSet_Ioi
   rw [← integral_indicator (this a), ← integral_indicator (this (b * a)),
-    ← Measure.integral_comp_mul_left]
+    ← abs_of_pos (inv_pos.mpr hb), ← Measure.integral_comp_mul_left]
   congr
   ext1 x
   rw [← indicator_comp_right, preimage_const_mul_Ioi _ hb, mul_div_cancel_left _ hb.ne']
@@ -989,7 +989,7 @@ theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 <
 #align measure_theory.integral_comp_mul_left_Ioi MeasureTheory.integral_comp_mul_left_Ioi
 
 theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
-    (∫ x in Ioi a, g (x * b)) = |b⁻¹| • ∫ x in Ioi (a * b), g x := by
+    (∫ x in Ioi a, g (x * b)) = b⁻¹ • ∫ x in Ioi (a * b), g x := by
   simpa only [mul_comm] using integral_comp_mul_left_Ioi g a hb
 #align measure_theory.integral_comp_mul_right_Ioi MeasureTheory.integral_comp_mul_right_Ioi
 

This makes it possible to use Gaussians as peak functions in the proof of Fourier inversion.

@@ -124,6 +124,26 @@ theorem mono {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle
 
 end AECover
 
+section MetricSpace
+
+variable [PseudoMetricSpace α] [OpensMeasurableSpace α]
+
+theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
+    AECover μ l (fun i ↦ Metric.ball x (r i)) where
+  measurableSet _ := Metric.isOpen_ball.measurableSet
+  ae_eventually_mem := by
+    apply eventually_of_forall (fun y ↦ ?_)
+    filter_upwards [hr (Ioi_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
+
+theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
+    AECover μ l (fun i ↦ Metric.closedBall x (r i)) where
+  measurableSet _ := Metric.isClosed_ball.measurableSet
+  ae_eventually_mem := by
+    apply eventually_of_forall (fun y ↦ ?_)
+    filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
+
+end MetricSpace
+
 section Preorderα
 
 variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α]

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -861,6 +861,22 @@ theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
 
 end IicFTC
 
+section UnivFTC
+
+variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m n : E} [NormedAddCommGroup E]
+  [NormedSpace ℝ E] [CompleteSpace E]
+
+theorem integral_of_hasDerivAt_of_tendsto
+    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f')
+    (hbot : Tendsto f atBot (𝓝 m)) (htop : Tendsto f atTop (𝓝 n)) : ∫ x, f' x = n - m := by
+  rw [← integral_univ, ← Set.Iic_union_Ioi (a := 0),
+    integral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi hf'.integrableOn hf'.integrableOn,
+    integral_Iic_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn hbot,
+    integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn htop]
+  abel
+
+end UnivFTC
+
 section IoiChangeVariables
 
 open Real

example use case: cocompact_le with integrable_iff_integrableAtFilter_cocompact from #10248 becomes a way to prove integrability from big-O estimates (e.g. #10258)

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -707,7 +707,7 @@ theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
   rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub]
   refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
   rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
-  exact h2f.filter_mono atTop_le_cocompact |>.tendsto
+  exact h2f.filter_mono _root_.atTop_le_cocompact |>.tendsto
 
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
@@ -857,7 +857,7 @@ theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
   rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
   refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
   rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
-  exact h2f.filter_mono atBot_le_cocompact |>.tendsto
+  exact h2f.filter_mono _root_.atBot_le_cocompact |>.tendsto
 
 end IicFTC
 

This better matches other lemma names.

From LeanAPAP

@@ -932,7 +932,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
   rw [a3] at this; rw [this]
   refine' set_integral_congr measurableSet_Ioi _
   intro x hx; dsimp only
-  rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
+  rw [abs_mul, abs_of_nonneg (rpow_nonneg (le_of_lt hx) _)]
 #align measure_theory.integral_comp_rpow_Ioi MeasureTheory.integral_comp_rpow_Ioi
 
 theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
@@ -990,7 +990,7 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
   rw [a3] at this
   rw [this]
   refine' integrableOn_congr_fun (fun x hx => _) measurableSet_Ioi
-  simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]
+  simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg (le_of_lt hx) _)]
 #align measure_theory.integrable_on_Ioi_comp_rpow_iff MeasureTheory.integrableOn_Ioi_comp_rpow_iff
 
 /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability (version

Subset of #9319

@@ -341,7 +341,7 @@ theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AE
 -- porting note: generalized from `[SemilatticeSup ι] [Nonempty ι]` to `[Preorder ι]`
 theorem AECover.biInter_Ici_aecover [Preorder ι] {φ : ι → Set α}
     (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (_h : k ∈ Ici n), φ k where
-  ae_eventually_mem := hφ.ae_eventually_mem.mono <| fun x h ↦ by
+  ae_eventually_mem := hφ.ae_eventually_mem.mono fun x h ↦ by
     simpa only [mem_iInter, mem_Ici, eventually_forall_ge_atTop]
   measurableSet i := .biInter (to_countable _) fun n _ => hφ.measurableSet n
 #align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AECover.biInter_Ici_aecover

We have in the library the lemma not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter, saying that if a function tends to infinity at a point in an interval [a, b], then its derivative is not interval-integrable on [a, b]. We generalize this result to allow for any set instead of [a, b], and apply it to half-infinite intervals.

In particular, we characterize integrability of x^s on [a, +oo), and deduce that x^s is never integrable on [0, +oo). This makes it possible to remove one assumption in Lemma mellin_comp_rpow on the Mellin transform.

@@ -683,7 +683,7 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici
   apply
     intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self)
       fun y hy => hderiv y hy.1
-  rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x]
   exact f'int.mono (fun y hy => hy.1) le_rfl
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
 
@@ -731,7 +731,7 @@ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a
       symm
       apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x
         (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1
-      rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]
+      rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x]
       exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
         (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
     _ = ∫ y in a..id x, ‖g' y‖ := by
@@ -835,7 +835,7 @@ theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic
   symm
   apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hx
     (hcont.mono Icc_subset_Iic_self) fun y hy => hderiv y hy.2
-  rw [intervalIntegrable_iff_integrable_Ioc_of_le hx]
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hx]
   exact f'int.mono (fun y hy => hy.2) le_rfl
 
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
@@ -867,7 +867,7 @@ open Real
 
 open scoped Interval
 
-variable {E : Type*} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+variable {E : Type*} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E]
 
 /-- Change-of-variables formula for `Ioi` integrals of vector-valued functions, proved by taking
 limits from the result for finite intervals. -/

Fixes mistake introduced in #7976 (ping to @alreadydone that you should use @[to_additive (attr := simp)] and not @[to_additive, simp]) and some namespacing mistakes from my own PRs #7337 and #7755

@@ -701,8 +701,8 @@ theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDer
 
 /-- A special case of `integral_Ioi_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
 compact support. -/
-theorem HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f)
-    (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by
+theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
+    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by
   have := fun x (_ : x ∈ Ioi b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
   rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub]
   refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
@@ -851,8 +851,8 @@ theorem integral_Iic_of_hasDerivAt_of_tendsto'
 
 /-- A special case of `integral_Iic_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
 compact support. -/
-theorem HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f)
-    (b : ℝ) : ∫ x in Iic b, deriv f x = f b := by
+theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
+    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Iic b, deriv f x = f b := by
   have := fun x (_ : x ∈ Iio b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
   rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
   refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn

• The extra import doesn't import any additional files transitively.
• From the Sobolev project

Co-authored-by: Heather Macbeth 25316162+hrmacbeth@users.noreply.github.com

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -3,6 +3,7 @@ Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
 -/
+import Mathlib.Analysis.Calculus.Deriv.Support
 import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
 import Mathlib.MeasureTheory.Integral.FundThmCalculus
 import Mathlib.Order.Filter.AtTopBot
@@ -665,10 +666,15 @@ variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m :
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
 When a function has a limit at infinity `m`, and its derivative is integrable, then the
 integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
-on `(a, +∞)` and continuity on `[a, +∞)`.-/
-theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
+on `(a, +∞)` and continuity at `a⁺`.-/
+theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
     (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a := by
+  have hcont : ContinuousOn f (Ici a) := by
+    intro x hx
+    rcases hx.out.eq_or_lt with rfl|hx
+    · exact hcont
+    · exact (hderiv x hx).continuousAt.continuousWithinAt
   refine' tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi a f'int tendsto_id) _
   apply Tendsto.congr' _ (hf.sub_const _)
   filter_upwards [Ioi_mem_atTop a] with x hx
@@ -688,17 +694,32 @@ on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
     (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
     ∫ x in Ioi a, f' x = m - f a := by
-  refine integral_Ioi_of_hasDerivAt_of_tendsto (fun x hx ↦ ?_) (fun x hx => hderiv x hx.out.le)
+  refine integral_Ioi_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le)
     f'int hf
-  exact (hderiv x hx).continuousAt.continuousWithinAt
+  exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto' MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto'
 
+/-- A special case of `integral_Ioi_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
+compact support. -/
+theorem HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f)
+    (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by
+  have := fun x (_ : x ∈ Ioi b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
+  rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub]
+  refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
+  rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
+  exact h2f.filter_mono atTop_le_cocompact |>.tendsto
+
 /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
-on `(a, +∞)` and continuity on `[a, +∞)`. -/
-theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousOn g (Ici a))
+on `(a, +∞)` and continuity at `a⁺`. -/
+theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
     (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by
+  have hcont : ContinuousOn g (Ici a) := by
+    intro x hx
+    rcases hx.out.eq_or_lt with rfl|hx
+    · exact hcont
+    · exact (hderiv x hx).continuousAt.continuousWithinAt
   refine integrableOn_Ioi_of_intervalIntegral_norm_tendsto (l - g a) a (fun x => ?_) tendsto_id ?_
   · exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
       (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1
@@ -726,15 +747,15 @@ is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
 theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by
-  refine integrableOn_Ioi_deriv_of_nonneg (fun x hx ↦ ?_) (fun x hx => hderiv x hx.out.le) g'pos hg
-  exact (hderiv x hx).continuousAt.continuousWithinAt
+  refine integrableOn_Ioi_deriv_of_nonneg ?_ (fun x hx => hderiv x hx.out.le) g'pos hg
+  exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt
 #align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOn_Ioi_deriv_of_nonneg'
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
-continuity on `[a, +∞)`. -/
-theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
+continuity at `a⁺`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
     (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
@@ -752,8 +773,8 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDeri
 
 /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
 is automatically integrable on `(a, +∞)`. Version assuming differentiability
-on `(a, +∞)` and continuity on `[a, +∞)`. -/
-theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousOn g (Ici a))
+on `(a, +∞)` and continuity at `a⁺`. -/
+theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
     (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by
   apply integrable_neg_iff.1
@@ -766,15 +787,15 @@ is automatically integrable on `(a, +∞)`. Version assuming differentiability
 on `[a, +∞)`. -/
 theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
     (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by
-  refine integrableOn_Ioi_deriv_of_nonpos (fun x hx ↦ ?_) (fun x hx ↦ hderiv x hx.out.le) g'neg hg
-  exact (hderiv x hx).continuousAt.continuousWithinAt
+  refine integrableOn_Ioi_deriv_of_nonpos ?_ (fun x hx ↦ hderiv x hx.out.le) g'neg hg
+  exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt
 #align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
 
 /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
-continuity on `[a, +∞)`. -/
-theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
+continuity at `a⁺`. -/
+theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
     (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
@@ -792,6 +813,54 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDeri
 
 end IoiFTC
 
+section IicFTC
+
+variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
+  [NormedSpace ℝ E] [CompleteSpace E]
+/-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
+When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
+integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
+on `(-∞, a)` and continuity at `a⁻`. -/
+theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
+    (hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
+    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by
+  have hcont : ContinuousOn f (Iic a) := by
+    intro x hx
+    rcases hx.out.eq_or_lt with rfl|hx
+    · exact hcont
+    · exact (hderiv x hx).continuousAt.continuousWithinAt
+  refine' tendsto_nhds_unique (intervalIntegral_tendsto_integral_Iic a f'int tendsto_id) _
+  apply Tendsto.congr' _ (hf.const_sub _)
+  filter_upwards [Iic_mem_atBot a] with x hx
+  symm
+  apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hx
+    (hcont.mono Icc_subset_Iic_self) fun y hy => hderiv y hy.2
+  rw [intervalIntegrable_iff_integrable_Ioc_of_le hx]
+  exact f'int.mono (fun y hy => hy.2) le_rfl
+
+/-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
+When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
+integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
+on `(-∞, a]`. -/
+theorem integral_Iic_of_hasDerivAt_of_tendsto'
+    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
+    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by
+  refine integral_Iic_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le)
+    f'int hf
+  exact (hderiv a right_mem_Iic).continuousAt.continuousWithinAt
+
+/-- A special case of `integral_Iic_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
+compact support. -/
+theorem HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f)
+    (b : ℝ) : ∫ x in Iic b, deriv f x = f b := by
+  have := fun x (_ : x ∈ Iio b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
+  rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
+  refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
+  rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
+  exact h2f.filter_mono atBot_le_cocompact |>.tendsto
+
+end IicFTC
+
 section IoiChangeVariables
 
 open Real

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

@@ -392,7 +392,7 @@ theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot]
   refine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt
     (fun i => lintegral_mono' Measure.restrict_le_self le_rfl) fun w hw => _
   rcases exists_between hw with ⟨m, hm₁, hm₂⟩
-  rcases(eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
+  rcases (eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
   exact ⟨i, lt_of_lt_of_le hm₁ hi⟩
 #align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated
 

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).

@@ -944,7 +944,7 @@ theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (
 
 theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
     IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) := by
-  simpa only [mul_comm, MulZeroClass.mul_zero] using integrableOn_Ioi_comp_mul_left_iff f c ha
+  simpa only [mul_comm, mul_zero] using integrableOn_Ioi_comp_mul_left_iff f c ha
 #align measure_theory.integrable_on_Ioi_comp_mul_right_iff MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
 
 end IoiIntegrability

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

This has nice performance benefits.

@@ -77,7 +77,7 @@ namespace MeasureTheory
 
 section AECover
 
-variable {α ι : Type _} [MeasurableSpace α] (μ : Measure α) (l : Filter ι)
+variable {α ι : Type*} [MeasurableSpace α] (μ : Measure α) (l : Filter ι)
 
 /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter
     `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if
@@ -292,14 +292,14 @@ theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s :
     (hφ.measurableSet i).inter hs
 #align measure_theory.ae_cover.inter_restrict MeasureTheory.AECover.inter_restrict
 
-theorem AECover.ae_tendsto_indicator {β : Type _} [Zero β] [TopologicalSpace β] (f : α → β)
+theorem AECover.ae_tendsto_indicator {β : Type*} [Zero β] [TopologicalSpace β] (f : α → β)
     {φ : ι → Set α} (hφ : AECover μ l φ) :
     ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <| f x) :=
   hφ.ae_eventually_mem.mono fun _x hx =>
     tendsto_const_nhds.congr' <| hx.mono fun _n hn => (indicator_of_mem hn _).symm
 #align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AECover.ae_tendsto_indicator
 
-theorem AECover.aemeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot]
+theorem AECover.aemeasurable {β : Type*} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot]
     {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
     (hfm : ∀ i, AEMeasurable f (μ.restrict <| φ i)) : AEMeasurable f μ := by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
@@ -309,7 +309,7 @@ theorem AECover.aemeasurable {β : Type _} [MeasurableSpace β] [l.IsCountablyGe
     mem_iUnion.mpr (hu.eventually hx).exists
 #align measure_theory.ae_cover.ae_measurable MeasureTheory.AECover.aemeasurable
 
-theorem AECover.aestronglyMeasurable {β : Type _} [TopologicalSpace β] [PseudoMetrizableSpace β]
+theorem AECover.aestronglyMeasurable {β : Type*} [TopologicalSpace β] [PseudoMetrizableSpace β]
     [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
     (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <| φ i)) : AEStronglyMeasurable f μ := by
   obtain ⟨u, hu⟩ := l.exists_seq_tendsto
@@ -320,7 +320,7 @@ theorem AECover.aestronglyMeasurable {β : Type _} [TopologicalSpace β] [Pseudo
 
 end AECover
 
-theorem AECover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
+theorem AECover.comp_tendsto {α ι ι' : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
     {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
     AECover μ l' (φ ∘ u) where
   ae_eventually_mem := hφ.ae_eventually_mem.mono fun _x hx => hu.eventually hx
@@ -329,7 +329,7 @@ theorem AECover.comp_tendsto {α ι ι' : Type _} [MeasurableSpace α] {μ : Mea
 
 section AECoverUnionInterCountable
 
-variable {α ι : Type _} [Countable ι] [MeasurableSpace α] {μ : Measure α}
+variable {α ι : Type*} [Countable ι] [MeasurableSpace α] {μ : Measure α}
 
 theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) :
     AECover μ atTop fun n : ι => ⋃ (k) (_h : k ∈ Iic n), φ k :=
@@ -349,7 +349,7 @@ end AECoverUnionInterCountable
 
 section Lintegral
 
-variable {α ι : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
+variable {α ι : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
 
 private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ)
     (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
@@ -400,7 +400,7 @@ end Lintegral
 
 section Integrable
 
-variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
+variable {α ι E : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
 theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
@@ -477,7 +477,7 @@ end Integrable
 
 section Integral
 
-variable {α ι E : Type _} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
+variable {α ι E : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
   [NormedSpace ℝ E] [CompleteSpace E]
 
 theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
@@ -511,7 +511,7 @@ end Integral
 
 section IntegrableOfIntervalIntegral
 
-variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l]
+variable {ι E : Type*} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l]
   [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E}
 
 theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
@@ -615,7 +615,7 @@ end IntegrableOfIntervalIntegral
 
 section IntegralOfIntervalIntegral
 
-variable {ι E : Type _} {μ : Measure ℝ} {l : Filter ι} [IsCountablyGenerated l]
+variable {ι E : Type*} {μ : Measure ℝ} {l : Filter ι} [IsCountablyGenerated l]
   [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {a b : ι → ℝ} {f : ℝ → E}
 
 theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot)
@@ -659,7 +659,7 @@ open scoped Interval
 
 section IoiFTC
 
-variable {E : Type _} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
+variable {E : Type*} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E]
   [NormedSpace ℝ E] [CompleteSpace E]
 
 /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
@@ -798,7 +798,7 @@ open Real
 
 open scoped Interval
 
-variable {E : Type _} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+variable {E : Type*} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
 /-- Change-of-variables formula for `Ioi` integrals of vector-valued functions, proved by taking
 limits from the result for finite intervals. -/
@@ -896,7 +896,7 @@ open Real
 
 open scoped Interval
 
-variable {E : Type _} [NormedAddCommGroup E]
+variable {E : Type*} [NormedAddCommGroup E]
 
 /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability. -/
 theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Bhavik Mehta
-
-! This file was ported from Lean 3 source module measure_theory.integral.integral_eq_improper
-! leanprover-community/mathlib commit b84aee748341da06a6d78491367e2c0e9f15e8a5
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
 import Mathlib.MeasureTheory.Integral.FundThmCalculus
@@ -14,6 +9,8 @@ import Mathlib.Order.Filter.AtTopBot
 import Mathlib.MeasureTheory.Function.Jacobian
 import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
 
+#align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
+
 /-!
 # Links between an integral and its "improper" version
 

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

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

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

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

@@ -863,7 +863,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     · intro hx; refine' ⟨x ^ (1 / p), rpow_pos_of_pos hx _, _⟩
       rw [← rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]
   have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g
-  rw [a3] at this ; rw [this]
+  rw [a3] at this; rw [this]
   refine' set_integral_congr measurableSet_Ioi _
   intro x hx; dsimp only
   rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]

@@ -390,7 +390,7 @@ theorem AECover.lintegral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ :
 
 theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot]
     [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞}
-    (hfm : AEMeasurable f μ) : (⨆ i : ι, ∫⁻ x in φ i, f x ∂μ) = ∫⁻ x, f x ∂μ := by
+    (hfm : AEMeasurable f μ) : ⨆ i : ι, ∫⁻ x in φ i, f x ∂μ = ∫⁻ x, f x ∂μ := by
   have := hφ.lintegral_tendsto_of_countably_generated hfm
   refine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt
     (fun i => lintegral_mono' Measure.restrict_le_self le_rfl) fun w hw => _

@@ -384,7 +384,7 @@ theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated
 
 theorem AECover.lintegral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
-    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : (∫⁻ x, f x ∂μ) = I :=
+    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto
 #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto
 
@@ -498,14 +498,14 @@ theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated]
     `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/
 theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
     (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
-    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : (∫ x, f x ∂μ) = I :=
+    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I :=
   tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h
 #align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto
 
 theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
     (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
-    (∫ x, f x ∂μ) = I :=
+    ∫ x, f x ∂μ = I :=
   have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto
   hφ.integral_eq_of_tendsto I hfi' htendsto
 #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
@@ -671,7 +671,7 @@ integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differe
 on `(a, +∞)` and continuity on `[a, +∞)`.-/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousOn f (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
-    (hf : Tendsto f atTop (𝓝 m)) : (∫ x in Ioi a, f' x) = m - f a := by
+    (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a := by
   refine' tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi a f'int tendsto_id) _
   apply Tendsto.congr' _ (hf.sub_const _)
   filter_upwards [Ioi_mem_atTop a] with x hx
@@ -690,7 +690,7 @@ integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differe
 on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
     (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
-    (∫ x in Ioi a, f' x) = m - f a := by
+    ∫ x in Ioi a, f' x = m - f a := by
   refine integral_Ioi_of_hasDerivAt_of_tendsto (fun x hx ↦ ?_) (fun x hx => hderiv x hx.out.le)
     f'int hf
   exact (hderiv x hx).continuousAt.continuousWithinAt
@@ -739,7 +739,7 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 continuity on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
-    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
@@ -748,7 +748,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousOn g (Ici a))
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
-    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonneg' hderiv g'pos hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
@@ -779,7 +779,7 @@ integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is in
 continuity on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
     (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
-    (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
     (integrableOn_Ioi_deriv_of_nonpos hcont hderiv g'neg hg) hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
@@ -788,7 +788,7 @@ theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousOn g (Ici a))
 integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
 `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
 theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
-    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : (∫ x in Ioi a, g' x) = l - g a :=
+    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a :=
   integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg)
     hg
 #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'

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

@@ -42,7 +42,7 @@ When using this definition with a measure restricted to a set `s`, which happens
 should not try too hard to use a `MeasureTheory.AECover` of subsets of `s`, as it often makes proofs
 more complicated than necessary. See for example the proof of
 `MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(λ x, Ioi x)` as a
-`MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(λ x, Ioc x b)`.
+`MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(fun x ↦ Ioc x b)`.
 
 ## Main statements
 

Part 2 of #5001

@@ -41,7 +41,7 @@ in φ i, f x ∂μ` as `i` tends to `l`.
 When using this definition with a measure restricted to a set `s`, which happens fairly often, one
 should not try too hard to use a `MeasureTheory.AECover` of subsets of `s`, as it often makes proofs
 more complicated than necessary. See for example the proof of
-`MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(λ x, Ioi x)` as an
+`MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(λ x, Ioi x)` as a
 `MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(λ x, Ioc x b)`.
 
 ## Main statements