probability.kernel.measurable_integral ⟷ Mathlib.Probability.Kernel.MeasurableIntegral

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

@@ -51,7 +51,7 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     simp only [preimage_empty, measure_empty, measurable_const]
   · -- case of a box: `t = t₁ ×ˢ t₂` for measurable sets `t₁` and `t₂`
     intro t' ht'
-    simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht' 
+    simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht'
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
     classical
     simp_rw [mk_preimage_prod_right_eq_if]
@@ -164,7 +164,7 @@ theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (h
   by
   let F : ℕ → simple_func (α × β) ℝ≥0∞ := simple_func.eapprox (uncurry f)
   have h : ∀ a, (⨆ n, F n a) = uncurry f a := simple_func.supr_eapprox_apply (uncurry f) hf
-  simp only [Prod.forall, uncurry_apply_pair] at h 
+  simp only [Prod.forall, uncurry_apply_pair] at h
   simp_rw [← h]
   have : ∀ a, ∫⁻ b, ⨆ n, F n (a, b) ∂κ a = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a :=
     by
@@ -327,7 +327,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
 
 #print MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' /-
 theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
-    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf ;
+    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf;
   exact hf.integral_kernel_prod_right
 #align measure_theory.strongly_measurable.integral_kernel_prod_right' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
 -/

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

@@ -54,6 +54,14 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht' 
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
     classical
+    simp_rw [mk_preimage_prod_right_eq_if]
+    have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
+      by
+      ext1 a
+      split_ifs
+      exacts [rfl, measure_empty]
+    rw [h_eq_ite]
+    exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
   · -- we assume that the result is true for `t` and we prove it for `tᶜ`
     intro t' ht' h_meas
     have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' :=
@@ -274,6 +282,46 @@ variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
   classical
+  borelize E
+  haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
+    hf.separable_space_range_union_singleton
+  let s : ℕ → simple_func (α × β) E :=
+    simple_func.approx_on _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
+  let s' : ℕ → α → simple_func β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
+  let f' : ℕ → α → E := fun n =>
+    {x | integrable (f x) (κ x)}.indicator fun x => (s' n x).integral (κ x)
+  have hf' : ∀ n, strongly_measurable (f' n) :=
+    by
+    intro n; refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
+    have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
+      by
+      intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
+      simp_rw [simple_func.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
+    simp only [simple_func.integral_eq_sum_of_subset (this _)]
+    refine' Finset.stronglyMeasurable_sum _ fun x _ => _
+    refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
+    simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
+    apply measurable_kernel_prod_mk_left
+    exact (s n).measurableSet_fiber x
+  have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
+    by
+    rw [tendsto_pi_nhds]; intro x
+    by_cases hfx : integrable (f x) (κ x)
+    · have : ∀ n, integrable (s' n x) (κ x) := by
+        intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
+        apply eventually_of_forall; intro y
+        simp_rw [s', simple_func.coe_comp]; exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
+      simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
+        mem_set_of_eq]
+      refine'
+        tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
+          (fun n => (s' n x).AEStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
+      · exact fun n => eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n
+      · refine' eventually_of_forall fun y => simple_func.tendsto_approx_on _ _ _
+        apply subset_closure
+        simp [-uncurry_apply_pair]
+    · simp [f', hfx, integral_undef]
+  exact stronglyMeasurable_of_tendsto _ hf' h2f'
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 -/
 

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

@@ -54,14 +54,6 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht' 
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
     classical
-    simp_rw [mk_preimage_prod_right_eq_if]
-    have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
-      by
-      ext1 a
-      split_ifs
-      exacts [rfl, measure_empty]
-    rw [h_eq_ite]
-    exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
   · -- we assume that the result is true for `t` and we prove it for `tᶜ`
     intro t' ht' h_meas
     have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' :=
@@ -282,46 +274,6 @@ variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
   classical
-  borelize E
-  haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
-    hf.separable_space_range_union_singleton
-  let s : ℕ → simple_func (α × β) E :=
-    simple_func.approx_on _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
-  let s' : ℕ → α → simple_func β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
-  let f' : ℕ → α → E := fun n =>
-    {x | integrable (f x) (κ x)}.indicator fun x => (s' n x).integral (κ x)
-  have hf' : ∀ n, strongly_measurable (f' n) :=
-    by
-    intro n; refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
-    have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
-      by
-      intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
-      simp_rw [simple_func.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
-    simp only [simple_func.integral_eq_sum_of_subset (this _)]
-    refine' Finset.stronglyMeasurable_sum _ fun x _ => _
-    refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
-    simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
-    apply measurable_kernel_prod_mk_left
-    exact (s n).measurableSet_fiber x
-  have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
-    by
-    rw [tendsto_pi_nhds]; intro x
-    by_cases hfx : integrable (f x) (κ x)
-    · have : ∀ n, integrable (s' n x) (κ x) := by
-        intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
-        apply eventually_of_forall; intro y
-        simp_rw [s', simple_func.coe_comp]; exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
-      simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
-        mem_set_of_eq]
-      refine'
-        tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
-          (fun n => (s' n x).AEStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
-      · exact fun n => eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n
-      · refine' eventually_of_forall fun y => simple_func.tendsto_approx_on _ _ _
-        apply subset_closure
-        simp [-uncurry_apply_pair]
-    · simp [f', hfx, integral_undef]
-  exact stronglyMeasurable_of_tendsto _ hf' h2f'
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 -/
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathbin.Probability.Kernel.Basic
+import Probability.Kernel.Basic
 
 #align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
 

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

@@ -2,14 +2,11 @@
 Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.measurable_integral
-! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Kernel.Basic
 
+#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
+
 /-!
 # Measurability of the integral against a kernel
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -42,7 +42,7 @@ namespace ProbabilityTheory
 
 namespace Kernel
 
-#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite /-
+#print ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite /-
 /-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
 theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
     (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
@@ -105,10 +105,10 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) _ (hf_meas i)
     rw [h_tsum]
     exact Measurable.ennreal_tsum hf
-#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
+#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite
 -/
 
-#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left /-
+#print ProbabilityTheory.kernel.measurable_kernel_prod_mk_left /-
 theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
     (ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
   by
@@ -118,10 +118,10 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
   simp_rw [this]
   refine' Measurable.ennreal_tsum fun n => _
   exact measurable_kernel_prod_mk_left_of_finite ht inferInstance
-#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left
+#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.kernel.measurable_kernel_prod_mk_left
 -/
 
-#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left' /-
+#print ProbabilityTheory.kernel.measurable_kernel_prod_mk_left' /-
 theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
     (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) :=
   by
@@ -130,14 +130,14 @@ theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)
   simp_rw [this]
   refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
   exact (measurable_fst.snd.prod_mk measurable_snd) hs
-#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'
+#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'
 -/
 
-#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right /-
+#print ProbabilityTheory.kernel.measurable_kernel_prod_mk_right /-
 theorem measurable_kernel_prod_mk_right [IsSFiniteKernel κ] {s : Set (β × α)}
     (hs : MeasurableSet s) : Measurable fun y => κ y ((fun x => (x, y)) ⁻¹' s) :=
   measurable_kernel_prod_mk_left (measurableSet_swap_iff.mpr hs)
-#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right
+#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.kernel.measurable_kernel_prod_mk_right
 -/
 
 end Kernel
@@ -148,14 +148,14 @@ section Lintegral
 
 variable [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
-#print ProbabilityTheory.Kernel.measurable_lintegral_indicator_const /-
+#print ProbabilityTheory.kernel.measurable_lintegral_indicator_const /-
 /-- Auxiliary lemma for `measurable.lintegral_kernel_prod_right`. -/
-theorem Kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
+theorem kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
     (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a :=
   by
   simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
   exact Measurable.const_mul (measurable_kernel_prod_mk_left ht) c
-#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.Kernel.measurable_lintegral_indicator_const
+#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.kernel.measurable_lintegral_indicator_const
 -/
 
 #print Measurable.lintegral_kernel_prod_right /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.measurable_integral
-! leanprover-community/mathlib commit 28b2a92f2996d28e580450863c130955de0ed398
+! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Probability.Kernel.Basic
 /-!
 # Measurability of the integral against a kernel
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The Lebesgue integral of a measurable function against a kernel is measurable. The Bochner integral
 is strongly measurable.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8efcf8022aac8e01df8d302dcebdbc25d6a886c8

@@ -39,6 +39,7 @@ namespace ProbabilityTheory
 
 namespace Kernel
 
+#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite /-
 /-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
 theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
     (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
@@ -101,8 +102,10 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) _ (hf_meas i)
     rw [h_tsum]
     exact Measurable.ennreal_tsum hf
-#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite
+#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
+-/
 
+#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left /-
 theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
     (ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
   by
@@ -112,8 +115,10 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
   simp_rw [this]
   refine' Measurable.ennreal_tsum fun n => _
   exact measurable_kernel_prod_mk_left_of_finite ht inferInstance
-#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.kernel.measurable_kernel_prod_mk_left
+#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left
+-/
 
+#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left' /-
 theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
     (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) :=
   by
@@ -122,12 +127,15 @@ theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)
   simp_rw [this]
   refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
   exact (measurable_fst.snd.prod_mk measurable_snd) hs
-#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'
+#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'
+-/
 
+#print ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right /-
 theorem measurable_kernel_prod_mk_right [IsSFiniteKernel κ] {s : Set (β × α)}
     (hs : MeasurableSet s) : Measurable fun y => κ y ((fun x => (x, y)) ⁻¹' s) :=
   measurable_kernel_prod_mk_left (measurableSet_swap_iff.mpr hs)
-#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.kernel.measurable_kernel_prod_mk_right
+#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right
+-/
 
 end Kernel
 
@@ -137,14 +145,17 @@ section Lintegral
 
 variable [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
+#print ProbabilityTheory.Kernel.measurable_lintegral_indicator_const /-
 /-- Auxiliary lemma for `measurable.lintegral_kernel_prod_right`. -/
-theorem kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
+theorem Kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
     (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a :=
   by
   simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
   exact Measurable.const_mul (measurable_kernel_prod_mk_left ht) c
-#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.kernel.measurable_lintegral_indicator_const
+#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.Kernel.measurable_lintegral_indicator_const
+-/
 
+#print Measurable.lintegral_kernel_prod_right /-
 /-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is measurable when seen as a
 map from `α × β` (hypothesis `measurable (uncurry f)`), the integral `a ↦ ∫⁻ b, f a b ∂(κ a)` is
 measurable. -/
@@ -178,7 +189,9 @@ theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (h
     rw [h_add]
     exact Measurable.add hm₁ hm₂
 #align measurable.lintegral_kernel_prod_right Measurable.lintegral_kernel_prod_right
+-/
 
+#print Measurable.lintegral_kernel_prod_right' /-
 theorem Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0∞} (hf : Measurable f) :
     Measurable fun a => ∫⁻ b, f (a, b) ∂κ a :=
   by
@@ -187,7 +200,9 @@ theorem Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0∞} (h
     rw [← @Prod.mk.eta _ _ x, uncurry_apply_pair]
   rwa [this]
 #align measurable.lintegral_kernel_prod_right' Measurable.lintegral_kernel_prod_right'
+-/
 
+#print Measurable.lintegral_kernel_prod_right'' /-
 theorem Measurable.lintegral_kernel_prod_right'' {f : β × γ → ℝ≥0∞} (hf : Measurable f) :
     Measurable fun x => ∫⁻ y, f (x, y) ∂η (a, x) :=
   by
@@ -197,47 +212,62 @@ theorem Measurable.lintegral_kernel_prod_right'' {f : β × γ → ℝ≥0∞} (
   refine' (Measurable.lintegral_kernel_prod_right' _).comp measurable_prod_mk_left
   exact hf.comp (measurable_fst.snd.prod_mk measurable_snd)
 #align measurable.lintegral_kernel_prod_right'' Measurable.lintegral_kernel_prod_right''
+-/
 
+#print Measurable.set_lintegral_kernel_prod_right /-
 theorem Measurable.set_lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (hf : Measurable (uncurry f))
     {s : Set β} (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f a b ∂κ a := by
   simp_rw [← lintegral_restrict κ hs]; exact hf.lintegral_kernel_prod_right
 #align measurable.set_lintegral_kernel_prod_right Measurable.set_lintegral_kernel_prod_right
+-/
 
+#print Measurable.lintegral_kernel_prod_left' /-
 theorem Measurable.lintegral_kernel_prod_left' {f : β × α → ℝ≥0∞} (hf : Measurable f) :
     Measurable fun y => ∫⁻ x, f (x, y) ∂κ y :=
   (measurable_swap_iff.mpr hf).lintegral_kernel_prod_right'
 #align measurable.lintegral_kernel_prod_left' Measurable.lintegral_kernel_prod_left'
+-/
 
+#print Measurable.lintegral_kernel_prod_left /-
 theorem Measurable.lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf : Measurable (uncurry f)) :
     Measurable fun y => ∫⁻ x, f x y ∂κ y :=
   hf.lintegral_kernel_prod_left'
 #align measurable.lintegral_kernel_prod_left Measurable.lintegral_kernel_prod_left
+-/
 
+#print Measurable.set_lintegral_kernel_prod_left /-
 theorem Measurable.set_lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf : Measurable (uncurry f))
     {s : Set β} (hs : MeasurableSet s) : Measurable fun b => ∫⁻ a in s, f a b ∂κ b := by
   simp_rw [← lintegral_restrict κ hs]; exact hf.lintegral_kernel_prod_left
 #align measurable.set_lintegral_kernel_prod_left Measurable.set_lintegral_kernel_prod_left
+-/
 
+#print Measurable.lintegral_kernel /-
 theorem Measurable.lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) :
     Measurable fun a => ∫⁻ b, f b ∂κ a :=
   Measurable.lintegral_kernel_prod_right (hf.comp measurable_snd)
 #align measurable.lintegral_kernel Measurable.lintegral_kernel
+-/
 
+#print Measurable.set_lintegral_kernel /-
 theorem Measurable.set_lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) {s : Set β}
     (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f b ∂κ a :=
   Measurable.set_lintegral_kernel_prod_right (hf.comp measurable_snd) hs
 #align measurable.set_lintegral_kernel Measurable.set_lintegral_kernel
+-/
 
 end Lintegral
 
 variable {E : Type _} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
+#print ProbabilityTheory.measurableSet_kernel_integrable /-
 theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
     MeasurableSet {x | Integrable (f x) (κ x)} :=
   by
   simp_rw [integrable, hf.of_uncurry_left.ae_strongly_measurable, true_and_iff]
   exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
 #align probability_theory.measurable_set_kernel_integrable ProbabilityTheory.measurableSet_kernel_integrable
+-/
 
 end ProbabilityTheory
 
@@ -248,6 +278,7 @@ namespace MeasureTheory
 variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [IsSFiniteKernel κ]
   [IsSFiniteKernel η]
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_right /-
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
   classical
@@ -292,12 +323,16 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     · simp [f', hfx, integral_undef]
   exact stronglyMeasurable_of_tendsto _ hf' h2f'
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
+-/
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' /-
 theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
     StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf ;
   exact hf.integral_kernel_prod_right
 #align measure_theory.strongly_measurable.integral_kernel_prod_right' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
+-/
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'' /-
 theorem StronglyMeasurable.integral_kernel_prod_right'' {f : β × γ → E}
     (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂η (a, x) :=
   by
@@ -308,17 +343,23 @@ theorem StronglyMeasurable.integral_kernel_prod_right'' {f : β × γ → E}
   refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' _
   exact hf.comp_measurable (measurable_fst.snd.prod_mk measurable_snd)
 #align measure_theory.strongly_measurable.integral_kernel_prod_right'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right''
+-/
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_left /-
 theorem StronglyMeasurable.integral_kernel_prod_left ⦃f : β → α → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂κ y :=
   (hf.comp_measurable measurable_swap).integral_kernel_prod_right'
 #align measure_theory.strongly_measurable.integral_kernel_prod_left MeasureTheory.StronglyMeasurable.integral_kernel_prod_left
+-/
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' /-
 theorem StronglyMeasurable.integral_kernel_prod_left' ⦃f : β × α → E⦄ (hf : StronglyMeasurable f) :
     StronglyMeasurable fun y => ∫ x, f (x, y) ∂κ y :=
   (hf.comp_measurable measurable_swap).integral_kernel_prod_right'
 #align measure_theory.strongly_measurable.integral_kernel_prod_left' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'
+-/
 
+#print MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'' /-
 theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf : StronglyMeasurable f) :
     StronglyMeasurable fun y => ∫ x, f (x, y) ∂η (a, y) :=
   by
@@ -329,6 +370,7 @@ theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf
   refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' _
   exact hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd)
 #align measure_theory.strongly_measurable.integral_kernel_prod_left'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
+-/
 
 end MeasureTheory
 

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

@@ -155,7 +155,7 @@ theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (h
   have h : ∀ a, (⨆ n, F n a) = uncurry f a := simple_func.supr_eapprox_apply (uncurry f) hf
   simp only [Prod.forall, uncurry_apply_pair] at h 
   simp_rw [← h]
-  have : ∀ a, (∫⁻ b, ⨆ n, F n (a, b) ∂κ a) = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a :=
+  have : ∀ a, ∫⁻ b, ⨆ n, F n (a, b) ∂κ a = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a :=
     by
     intro a
     rw [lintegral_supr]

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

@@ -41,7 +41,7 @@ namespace Kernel
 
 /-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
 theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
-    (hκs : ∀ a, FiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
+    (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
   by
   -- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
   -- by boxes to prove the result by induction.
@@ -53,14 +53,14 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht' 
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
     classical
-      simp_rw [mk_preimage_prod_right_eq_if]
-      have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
-        by
-        ext1 a
-        split_ifs
-        exacts [rfl, measure_empty]
-      rw [h_eq_ite]
-      exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
+    simp_rw [mk_preimage_prod_right_eq_if]
+    have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
+      by
+      ext1 a
+      split_ifs
+      exacts [rfl, measure_empty]
+    rw [h_eq_ite]
+    exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
   · -- we assume that the result is true for `t` and we prove it for `tᶜ`
     intro t' ht' h_meas
     have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' :=
@@ -117,7 +117,7 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
 theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
     (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) :=
   by
-  have : ∀ b, Prod.mk b ⁻¹' s = { c | ((a, b), c) ∈ { p : (α × β) × γ | (p.1.2, p.2) ∈ s } } := by
+  have : ∀ b, Prod.mk b ⁻¹' s = {c | ((a, b), c) ∈ {p : (α × β) × γ | (p.1.2, p.2) ∈ s}} := by
     intro b; rfl
   simp_rw [this]
   refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
@@ -233,7 +233,7 @@ end Lintegral
 variable {E : Type _} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
 theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
-    MeasurableSet { x | Integrable (f x) (κ x) } :=
+    MeasurableSet {x | Integrable (f x) (κ x)} :=
   by
   simp_rw [integrable, hf.of_uncurry_left.ae_strongly_measurable, true_and_iff]
   exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
@@ -251,48 +251,46 @@ variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
   classical
-    borelize E
-    haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
-      hf.separable_space_range_union_singleton
-    let s : ℕ → simple_func (α × β) E :=
-      simple_func.approx_on _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
-    let s' : ℕ → α → simple_func β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
-    let f' : ℕ → α → E := fun n =>
-      { x | integrable (f x) (κ x) }.indicator fun x => (s' n x).integral (κ x)
-    have hf' : ∀ n, strongly_measurable (f' n) :=
-      by
-      intro n; refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
-      have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
-        by
-        intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
-        simp_rw [simple_func.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
-      simp only [simple_func.integral_eq_sum_of_subset (this _)]
-      refine' Finset.stronglyMeasurable_sum _ fun x _ => _
-      refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
-      simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
-      apply measurable_kernel_prod_mk_left
-      exact (s n).measurableSet_fiber x
-    have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
+  borelize E
+  haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
+    hf.separable_space_range_union_singleton
+  let s : ℕ → simple_func (α × β) E :=
+    simple_func.approx_on _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
+  let s' : ℕ → α → simple_func β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
+  let f' : ℕ → α → E := fun n =>
+    {x | integrable (f x) (κ x)}.indicator fun x => (s' n x).integral (κ x)
+  have hf' : ∀ n, strongly_measurable (f' n) :=
+    by
+    intro n; refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
+    have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
       by
-      rw [tendsto_pi_nhds]; intro x
-      by_cases hfx : integrable (f x) (κ x)
-      · have : ∀ n, integrable (s' n x) (κ x) := by
-          intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
-          apply eventually_of_forall; intro y
-          simp_rw [s', simple_func.coe_comp]; exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
-        simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
-          mem_set_of_eq]
-        refine'
-          tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
-            (fun n => (s' n x).AEStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
-        ·
-          exact fun n =>
-            eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n
-        · refine' eventually_of_forall fun y => simple_func.tendsto_approx_on _ _ _
-          apply subset_closure
-          simp [-uncurry_apply_pair]
-      · simp [f', hfx, integral_undef]
-    exact stronglyMeasurable_of_tendsto _ hf' h2f'
+      intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
+      simp_rw [simple_func.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
+    simp only [simple_func.integral_eq_sum_of_subset (this _)]
+    refine' Finset.stronglyMeasurable_sum _ fun x _ => _
+    refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
+    simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
+    apply measurable_kernel_prod_mk_left
+    exact (s n).measurableSet_fiber x
+  have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
+    by
+    rw [tendsto_pi_nhds]; intro x
+    by_cases hfx : integrable (f x) (κ x)
+    · have : ∀ n, integrable (s' n x) (κ x) := by
+        intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
+        apply eventually_of_forall; intro y
+        simp_rw [s', simple_func.coe_comp]; exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
+      simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
+        mem_set_of_eq]
+      refine'
+        tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
+          (fun n => (s' n x).AEStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
+      · exact fun n => eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n
+      · refine' eventually_of_forall fun y => simple_func.tendsto_approx_on _ _ _
+        apply subset_closure
+        simp [-uncurry_apply_pair]
+    · simp [f', hfx, integral_undef]
+  exact stronglyMeasurable_of_tendsto _ hf' h2f'
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 
 theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :

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

@@ -50,7 +50,7 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     simp only [preimage_empty, measure_empty, measurable_const]
   · -- case of a box: `t = t₁ ×ˢ t₂` for measurable sets `t₁` and `t₂`
     intro t' ht'
-    simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht'
+    simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht' 
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
     classical
       simp_rw [mk_preimage_prod_right_eq_if]
@@ -58,7 +58,7 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
         by
         ext1 a
         split_ifs
-        exacts[rfl, measure_empty]
+        exacts [rfl, measure_empty]
       rw [h_eq_ite]
       exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
   · -- we assume that the result is true for `t` and we prove it for `tᶜ`
@@ -153,7 +153,7 @@ theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (h
   by
   let F : ℕ → simple_func (α × β) ℝ≥0∞ := simple_func.eapprox (uncurry f)
   have h : ∀ a, (⨆ n, F n a) = uncurry f a := simple_func.supr_eapprox_apply (uncurry f) hf
-  simp only [Prod.forall, uncurry_apply_pair] at h
+  simp only [Prod.forall, uncurry_apply_pair] at h 
   simp_rw [← h]
   have : ∀ a, (∫⁻ b, ⨆ n, F n (a, b) ∂κ a) = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a :=
     by
@@ -296,7 +296,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 
 theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
-    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf;
+    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf ;
   exact hf.integral_kernel_prod_right
 #align measure_theory.strongly_measurable.integral_kernel_prod_right' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
 

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

@@ -30,7 +30,7 @@ is strongly measurable.
 
 open MeasureTheory ProbabilityTheory Function Set Filter
 
-open MeasureTheory ENNReal Topology
+open scoped MeasureTheory ENNReal Topology
 
 variable {α β γ : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
   {κ : kernel α β} {η : kernel (α × β) γ} {a : α}

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

@@ -94,12 +94,8 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
       ext1 a
       rw [measure_Union]
       · intro i j hij s hsi hsj b hbs
-        have habi : {(a, b)} ⊆ f i := by
-          rw [Set.singleton_subset_iff]
-          exact hsi hbs
-        have habj : {(a, b)} ⊆ f j := by
-          rw [Set.singleton_subset_iff]
-          exact hsj hbs
+        have habi : {(a, b)} ⊆ f i := by rw [Set.singleton_subset_iff]; exact hsi hbs
+        have habj : {(a, b)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hbs
         simpa only [Set.bot_eq_empty, Set.le_eq_subset, Set.singleton_subset_iff,
           Set.mem_empty_iff_false] using h_disj hij habi habj
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) _ (hf_meas i)
@@ -121,10 +117,8 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
 theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
     (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) :=
   by
-  have : ∀ b, Prod.mk b ⁻¹' s = { c | ((a, b), c) ∈ { p : (α × β) × γ | (p.1.2, p.2) ∈ s } } :=
-    by
-    intro b
-    rfl
+  have : ∀ b, Prod.mk b ⁻¹' s = { c | ((a, b), c) ∈ { p : (α × β) × γ | (p.1.2, p.2) ∈ s } } := by
+    intro b; rfl
   simp_rw [this]
   refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
   exact (measurable_fst.snd.prod_mk measurable_snd) hs
@@ -189,9 +183,7 @@ theorem Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0∞} (h
     Measurable fun a => ∫⁻ b, f (a, b) ∂κ a :=
   by
   refine' Measurable.lintegral_kernel_prod_right _
-  have : (uncurry fun (a : α) (b : β) => f (a, b)) = f :=
-    by
-    ext x
+  have : (uncurry fun (a : α) (b : β) => f (a, b)) = f := by ext x;
     rw [← @Prod.mk.eta _ _ x, uncurry_apply_pair]
   rwa [this]
 #align measurable.lintegral_kernel_prod_right' Measurable.lintegral_kernel_prod_right'
@@ -207,10 +199,8 @@ theorem Measurable.lintegral_kernel_prod_right'' {f : β × γ → ℝ≥0∞} (
 #align measurable.lintegral_kernel_prod_right'' Measurable.lintegral_kernel_prod_right''
 
 theorem Measurable.set_lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (hf : Measurable (uncurry f))
-    {s : Set β} (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f a b ∂κ a :=
-  by
-  simp_rw [← lintegral_restrict κ hs]
-  exact hf.lintegral_kernel_prod_right
+    {s : Set β} (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f a b ∂κ a := by
+  simp_rw [← lintegral_restrict κ hs]; exact hf.lintegral_kernel_prod_right
 #align measurable.set_lintegral_kernel_prod_right Measurable.set_lintegral_kernel_prod_right
 
 theorem Measurable.lintegral_kernel_prod_left' {f : β × α → ℝ≥0∞} (hf : Measurable f) :
@@ -224,10 +214,8 @@ theorem Measurable.lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf
 #align measurable.lintegral_kernel_prod_left Measurable.lintegral_kernel_prod_left
 
 theorem Measurable.set_lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf : Measurable (uncurry f))
-    {s : Set β} (hs : MeasurableSet s) : Measurable fun b => ∫⁻ a in s, f a b ∂κ b :=
-  by
-  simp_rw [← lintegral_restrict κ hs]
-  exact hf.lintegral_kernel_prod_left
+    {s : Set β} (hs : MeasurableSet s) : Measurable fun b => ∫⁻ a in s, f a b ∂κ b := by
+  simp_rw [← lintegral_restrict κ hs]; exact hf.lintegral_kernel_prod_left
 #align measurable.set_lintegral_kernel_prod_left Measurable.set_lintegral_kernel_prod_left
 
 theorem Measurable.lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) :
@@ -273,16 +261,11 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
       { x | integrable (f x) (κ x) }.indicator fun x => (s' n x).integral (κ x)
     have hf' : ∀ n, strongly_measurable (f' n) :=
       by
-      intro n
-      refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
+      intro n; refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
       have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
         by
-        intro x
-        refine' Finset.Subset.trans (Finset.filter_subset _ _) _
-        intro y
-        simp_rw [simple_func.mem_range]
-        rintro ⟨z, rfl⟩
-        exact ⟨(x, z), rfl⟩
+        intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
+        simp_rw [simple_func.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
       simp only [simple_func.integral_eq_sum_of_subset (this _)]
       refine' Finset.stronglyMeasurable_sum _ fun x _ => _
       refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
@@ -291,16 +274,12 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
       exact (s n).measurableSet_fiber x
     have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
       by
-      rw [tendsto_pi_nhds]
-      intro x
+      rw [tendsto_pi_nhds]; intro x
       by_cases hfx : integrable (f x) (κ x)
       · have : ∀ n, integrable (s' n x) (κ x) := by
-          intro n
-          apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
-          apply eventually_of_forall
-          intro y
-          simp_rw [s', simple_func.coe_comp]
-          exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
+          intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
+          apply eventually_of_forall; intro y
+          simp_rw [s', simple_func.coe_comp]; exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
         simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
           mem_set_of_eq]
         refine'
@@ -317,9 +296,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 
 theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
-    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x :=
-  by
-  rw [← uncurry_curry f] at hf
+    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by rw [← uncurry_curry f] at hf;
   exact hf.integral_kernel_prod_right
 #align measure_theory.strongly_measurable.integral_kernel_prod_right' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
 

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

@@ -296,7 +296,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
       by_cases hfx : integrable (f x) (κ x)
       · have : ∀ n, integrable (s' n x) (κ x) := by
           intro n
-          apply (hfx.norm.add hfx.norm).mono' (s' n x).AeStronglyMeasurable
+          apply (hfx.norm.add hfx.norm).mono' (s' n x).AEStronglyMeasurable
           apply eventually_of_forall
           intro y
           simp_rw [s', simple_func.coe_comp]
@@ -305,7 +305,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
           mem_set_of_eq]
         refine'
           tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
-            (fun n => (s' n x).AeStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
+            (fun n => (s' n x).AEStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
         ·
           exact fun n =>
             eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n

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

@@ -104,7 +104,7 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
           Set.mem_empty_iff_false] using h_disj hij habi habj
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) _ (hf_meas i)
     rw [h_tsum]
-    exact Measurable.eNNReal_tsum hf
+    exact Measurable.ennreal_tsum hf
 #align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite
 
 theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
@@ -114,7 +114,7 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
   have : ∀ a, kernel.sum (seq κ) a (Prod.mk a ⁻¹' t) = ∑' n, seq κ n a (Prod.mk a ⁻¹' t) := fun a =>
     kernel.sum_apply' _ _ (measurable_prod_mk_left ht)
   simp_rw [this]
-  refine' Measurable.eNNReal_tsum fun n => _
+  refine' Measurable.ennreal_tsum fun n => _
   exact measurable_kernel_prod_mk_left_of_finite ht inferInstance
 #align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.kernel.measurable_kernel_prod_mk_left
 
@@ -285,7 +285,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
         exact ⟨(x, z), rfl⟩
       simp only [simple_func.integral_eq_sum_of_subset (this _)]
       refine' Finset.stronglyMeasurable_sum _ fun x _ => _
-      refine' (Measurable.eNNReal_toReal _).StronglyMeasurable.smul_const _
+      refine' (Measurable.ennreal_toReal _).StronglyMeasurable.smul_const _
       simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
       apply measurable_kernel_prod_mk_left
       exact (s n).measurableSet_fiber x

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

@@ -168,7 +168,7 @@ theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (h
     · exact fun n => (F n).Measurable.comp measurable_prod_mk_left
     · exact fun i j hij b => simple_func.monotone_eapprox (uncurry f) hij _
   simp_rw [this]
-  refine' measurable_supᵢ fun n => simple_func.induction _ _ (F n)
+  refine' measurable_iSup fun n => simple_func.induction _ _ (F n)
   · intro c t ht
     simp only [simple_func.const_zero, simple_func.coe_piecewise, simple_func.coe_const,
       simple_func.coe_zero, Set.piecewise_eq_indicator]

mathlib commit https://github.com/leanprover-community/mathlib/commit/28b2a92f2996d28e580450863c130955de0ed398

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.measurable_integral
-! leanprover-community/mathlib commit 483dd86cfec4a1380d22b1f6acd4c3dc53f501ff
+! leanprover-community/mathlib commit 28b2a92f2996d28e580450863c130955de0ed398
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,68 +13,68 @@ import Mathbin.Probability.Kernel.Basic
 /-!
 # Measurability of the integral against a kernel
 
-The Lebesgue integral of a measurable function against a kernel is measurable.
+The Lebesgue integral of a measurable function against a kernel is measurable. The Bochner integral
+is strongly measurable.
 
 ## Main statements
 
-* `probability_theory.kernel.measurable_lintegral`: the function `a ↦ ∫⁻ b, f a b ∂(κ a)` is
-  measurable, for an s-finite kernel `κ : kernel α β` and a function `f : α → β → ℝ≥0∞` such that
-  `function.uncurry f` is measurable.
-
+* `measurable.lintegral_kernel_prod_right`: the function `a ↦ ∫⁻ b, f a b ∂(κ a)` is measurable,
+  for an s-finite kernel `κ : kernel α β` and a function `f : α → β → ℝ≥0∞` such that `uncurry f`
+  is measurable.
+* `measure_theory.strongly_measurable.integral_kernel_prod_right`: the function
+  `a ↦ ∫ b, f a b ∂(κ a)` is measurable, for an s-finite kernel `κ : kernel α β` and a function
+  `f : α → β → E` such that `uncurry f` is measurable.
 
 -/
 
 
-open MeasureTheory ProbabilityTheory
+open MeasureTheory ProbabilityTheory Function Set Filter
 
-open MeasureTheory ENNReal NNReal BigOperators
+open MeasureTheory ENNReal Topology
 
-namespace ProbabilityTheory.kernel
+variable {α β γ : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
+  {κ : kernel α β} {η : kernel (α × β) γ} {a : α}
 
-variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+namespace ProbabilityTheory
 
-include mα mβ
+namespace Kernel
 
-/-- This is an auxiliary lemma for `measurable_prod_mk_mem`. -/
-theorem measurable_prod_mk_mem_of_finite (κ : kernel α β) {t : Set (α × β)} (ht : MeasurableSet t)
-    (hκs : ∀ a, FiniteMeasure (κ a)) : Measurable fun a => κ a { b | (a, b) ∈ t } :=
+/-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
+theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
+    (hκs : ∀ a, FiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
   by
   -- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
   -- by boxes to prove the result by induction.
   refine' MeasurableSpace.induction_on_inter generate_from_prod.symm isPiSystem_prod _ _ _ _ ht
   ·-- case `t = ∅`
-    simp only [Set.mem_empty_iff_false, Set.setOf_false, measure_empty, measurable_const]
+    simp only [preimage_empty, measure_empty, measurable_const]
   · -- case of a box: `t = t₁ ×ˢ t₂` for measurable sets `t₁` and `t₂`
     intro t' ht'
     simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht'
     obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
-    simp only [Set.prod_mk_mem_set_prod_eq]
     classical
-      have h_eq_ite :
-        (fun a => κ a { b : β | a ∈ t₁ ∧ b ∈ t₂ }) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
+      simp_rw [mk_preimage_prod_right_eq_if]
+      have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 :=
         by
         ext1 a
         split_ifs
-        · simp only [h, true_and_iff]
-          rfl
-        · simp only [h, false_and_iff, Set.setOf_false, Set.inter_empty, measure_empty]
+        exacts[rfl, measure_empty]
       rw [h_eq_ite]
       exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
   · -- we assume that the result is true for `t` and we prove it for `tᶜ`
     intro t' ht' h_meas
-    have h_eq_sdiff : ∀ a, { b : β | (a, b) ∈ t'ᶜ } = Set.univ \ { b : β | (a, b) ∈ t' } :=
+    have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' :=
       by
       intro a
       ext1 b
-      simp only [Set.mem_compl_iff, Set.mem_setOf_eq, Set.mem_diff, Set.mem_univ, true_and_iff]
+      simp only [mem_compl_iff, mem_preimage, mem_diff, mem_univ, true_and_iff]
     simp_rw [h_eq_sdiff]
     have :
-      (fun a => κ a (Set.univ \ { b : β | (a, b) ∈ t' })) = fun a =>
-        κ a Set.univ - κ a { b : β | (a, b) ∈ t' } :=
+      (fun a => κ a (Set.univ \ Prod.mk a ⁻¹' t')) = fun a =>
+        κ a Set.univ - κ a (Prod.mk a ⁻¹' t') :=
       by
       ext1 a
-      rw [← Set.diff_inter_self_eq_diff, Set.inter_univ, measure_diff]
-      · exact Set.subset_univ _
+      rw [← Set.diff_inter_self_eq_diff, Set.inter_univ, measure_diff (Set.subset_univ _)]
       · exact (@measurable_prod_mk_left α β _ _ a) t' ht'
       · exact measure_ne_top _ _
     rw [this]
@@ -82,16 +82,14 @@ theorem measurable_prod_mk_mem_of_finite (κ : kernel α β) {t : Set (α × β)
   · -- we assume that the result is true for a family of disjoint sets and prove it for their union
     intro f h_disj hf_meas hf
     have h_Union :
-      (fun a => κ a { b : β | (a, b) ∈ ⋃ i, f i }) = fun a => κ a (⋃ i, { b : β | (a, b) ∈ f i }) :=
+      (fun a => κ a (Prod.mk a ⁻¹' ⋃ i, f i)) = fun a => κ a (⋃ i, Prod.mk a ⁻¹' f i) :=
       by
       ext1 a
       congr with b
-      simp only [Set.mem_unionᵢ, Set.supᵢ_eq_unionᵢ, Set.mem_setOf_eq]
-      rfl
+      simp only [mem_Union, mem_preimage]
     rw [h_Union]
     have h_tsum :
-      (fun a => κ a (⋃ i, { b : β | (a, b) ∈ f i })) = fun a =>
-        ∑' i, κ a { b : β | (a, b) ∈ f i } :=
+      (fun a => κ a (⋃ i, Prod.mk a ⁻¹' f i)) = fun a => ∑' i, κ a (Prod.mk a ⁻¹' f i) :=
       by
       ext1 a
       rw [measure_Union]
@@ -107,51 +105,74 @@ theorem measurable_prod_mk_mem_of_finite (κ : kernel α β) {t : Set (α × β)
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) _ (hf_meas i)
     rw [h_tsum]
     exact Measurable.eNNReal_tsum hf
-#align probability_theory.kernel.measurable_prod_mk_mem_of_finite ProbabilityTheory.kernel.measurable_prod_mk_mem_of_finite
+#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite
 
-theorem measurable_prod_mk_mem (κ : kernel α β) [IsSFiniteKernel κ] {t : Set (α × β)}
-    (ht : MeasurableSet t) : Measurable fun a => κ a { b | (a, b) ∈ t } :=
+theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
+    (ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) :=
   by
   rw [← kernel_sum_seq κ]
-  have :
-    ∀ a, kernel.sum (seq κ) a { b : β | (a, b) ∈ t } = ∑' n, seq κ n a { b : β | (a, b) ∈ t } :=
-    fun a => kernel.sum_apply' _ _ (measurable_prod_mk_left ht)
+  have : ∀ a, kernel.sum (seq κ) a (Prod.mk a ⁻¹' t) = ∑' n, seq κ n a (Prod.mk a ⁻¹' t) := fun a =>
+    kernel.sum_apply' _ _ (measurable_prod_mk_left ht)
   simp_rw [this]
   refine' Measurable.eNNReal_tsum fun n => _
-  exact measurable_prod_mk_mem_of_finite (seq κ n) ht inferInstance
-#align probability_theory.kernel.measurable_prod_mk_mem ProbabilityTheory.kernel.measurable_prod_mk_mem
+  exact measurable_kernel_prod_mk_left_of_finite ht inferInstance
+#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.kernel.measurable_kernel_prod_mk_left
+
+theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
+    (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) :=
+  by
+  have : ∀ b, Prod.mk b ⁻¹' s = { c | ((a, b), c) ∈ { p : (α × β) × γ | (p.1.2, p.2) ∈ s } } :=
+    by
+    intro b
+    rfl
+  simp_rw [this]
+  refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
+  exact (measurable_fst.snd.prod_mk measurable_snd) hs
+#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'
+
+theorem measurable_kernel_prod_mk_right [IsSFiniteKernel κ] {s : Set (β × α)}
+    (hs : MeasurableSet s) : Measurable fun y => κ y ((fun x => (x, y)) ⁻¹' s) :=
+  measurable_kernel_prod_mk_left (measurableSet_swap_iff.mpr hs)
+#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.kernel.measurable_kernel_prod_mk_right
+
+end Kernel
+
+open ProbabilityTheory.kernel
+
+section Lintegral
+
+variable [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
-theorem measurable_lintegral_indicator_const (κ : kernel α β) [IsSFiniteKernel κ] {t : Set (α × β)}
-    (ht : MeasurableSet t) (c : ℝ≥0∞) :
-    Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a :=
+/-- Auxiliary lemma for `measurable.lintegral_kernel_prod_right`. -/
+theorem kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
+    (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a :=
   by
   simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
-  exact Measurable.const_mul (measurable_prod_mk_mem _ ht) c
+  exact Measurable.const_mul (measurable_kernel_prod_mk_left ht) c
 #align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.kernel.measurable_lintegral_indicator_const
 
 /-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is measurable when seen as a
-map from `α × β` (hypothesis `measurable (function.uncurry f)`), the integral
-`a ↦ ∫⁻ b, f a b ∂(κ a)` is measurable. -/
-theorem measurable_lintegral (κ : kernel α β) [IsSFiniteKernel κ] {f : α → β → ℝ≥0∞}
-    (hf : Measurable (Function.uncurry f)) : Measurable fun a => ∫⁻ b, f a b ∂κ a :=
+map from `α × β` (hypothesis `measurable (uncurry f)`), the integral `a ↦ ∫⁻ b, f a b ∂(κ a)` is
+measurable. -/
+theorem Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (hf : Measurable (uncurry f)) :
+    Measurable fun a => ∫⁻ b, f a b ∂κ a :=
   by
-  let F : ℕ → simple_func (α × β) ℝ≥0∞ := simple_func.eapprox (Function.uncurry f)
-  have h : ∀ a, (⨆ n, F n a) = Function.uncurry f a :=
-    simple_func.supr_eapprox_apply (Function.uncurry f) hf
-  simp only [Prod.forall, Function.uncurry_apply_pair] at h
+  let F : ℕ → simple_func (α × β) ℝ≥0∞ := simple_func.eapprox (uncurry f)
+  have h : ∀ a, (⨆ n, F n a) = uncurry f a := simple_func.supr_eapprox_apply (uncurry f) hf
+  simp only [Prod.forall, uncurry_apply_pair] at h
   simp_rw [← h]
   have : ∀ a, (∫⁻ b, ⨆ n, F n (a, b) ∂κ a) = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a :=
     by
     intro a
     rw [lintegral_supr]
     · exact fun n => (F n).Measurable.comp measurable_prod_mk_left
-    · exact fun i j hij b => simple_func.monotone_eapprox (Function.uncurry f) hij _
+    · exact fun i j hij b => simple_func.monotone_eapprox (uncurry f) hij _
   simp_rw [this]
   refine' measurable_supᵢ fun n => simple_func.induction _ _ (F n)
   · intro c t ht
     simp only [simple_func.const_zero, simple_func.coe_piecewise, simple_func.coe_const,
       simple_func.coe_zero, Set.piecewise_eq_indicator]
-    exact measurable_lintegral_indicator_const κ ht c
+    exact kernel.measurable_lintegral_indicator_const ht c
   · intro g₁ g₂ h_disj hm₁ hm₂
     simp only [simple_func.coe_add, Pi.add_apply]
     have h_add :
@@ -162,26 +183,177 @@ theorem measurable_lintegral (κ : kernel α β) [IsSFiniteKernel κ] {f : α 
       rw [Pi.add_apply, lintegral_add_left (g₁.measurable.comp measurable_prod_mk_left)]
     rw [h_add]
     exact Measurable.add hm₁ hm₂
-#align probability_theory.kernel.measurable_lintegral ProbabilityTheory.kernel.measurable_lintegral
+#align measurable.lintegral_kernel_prod_right Measurable.lintegral_kernel_prod_right
 
-theorem measurable_lintegral' (κ : kernel α β) [IsSFiniteKernel κ] {f : β → ℝ≥0∞}
-    (hf : Measurable f) : Measurable fun a => ∫⁻ b, f b ∂κ a :=
-  measurable_lintegral κ (hf.comp measurable_snd)
-#align probability_theory.kernel.measurable_lintegral' ProbabilityTheory.kernel.measurable_lintegral'
+theorem Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0∞} (hf : Measurable f) :
+    Measurable fun a => ∫⁻ b, f (a, b) ∂κ a :=
+  by
+  refine' Measurable.lintegral_kernel_prod_right _
+  have : (uncurry fun (a : α) (b : β) => f (a, b)) = f :=
+    by
+    ext x
+    rw [← @Prod.mk.eta _ _ x, uncurry_apply_pair]
+  rwa [this]
+#align measurable.lintegral_kernel_prod_right' Measurable.lintegral_kernel_prod_right'
+
+theorem Measurable.lintegral_kernel_prod_right'' {f : β × γ → ℝ≥0∞} (hf : Measurable f) :
+    Measurable fun x => ∫⁻ y, f (x, y) ∂η (a, x) :=
+  by
+  change
+    Measurable
+      ((fun x => ∫⁻ y, (fun u : (α × β) × γ => f (u.1.2, u.2)) (x, y) ∂η x) ∘ fun x => (a, x))
+  refine' (Measurable.lintegral_kernel_prod_right' _).comp measurable_prod_mk_left
+  exact hf.comp (measurable_fst.snd.prod_mk measurable_snd)
+#align measurable.lintegral_kernel_prod_right'' Measurable.lintegral_kernel_prod_right''
+
+theorem Measurable.set_lintegral_kernel_prod_right {f : α → β → ℝ≥0∞} (hf : Measurable (uncurry f))
+    {s : Set β} (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f a b ∂κ a :=
+  by
+  simp_rw [← lintegral_restrict κ hs]
+  exact hf.lintegral_kernel_prod_right
+#align measurable.set_lintegral_kernel_prod_right Measurable.set_lintegral_kernel_prod_right
+
+theorem Measurable.lintegral_kernel_prod_left' {f : β × α → ℝ≥0∞} (hf : Measurable f) :
+    Measurable fun y => ∫⁻ x, f (x, y) ∂κ y :=
+  (measurable_swap_iff.mpr hf).lintegral_kernel_prod_right'
+#align measurable.lintegral_kernel_prod_left' Measurable.lintegral_kernel_prod_left'
 
-theorem measurableSet_lintegral (κ : kernel α β) [IsSFiniteKernel κ] {f : α → β → ℝ≥0∞}
-    (hf : Measurable (Function.uncurry f)) {s : Set β} (hs : MeasurableSet s) :
-    Measurable fun a => ∫⁻ b in s, f a b ∂κ a :=
+theorem Measurable.lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf : Measurable (uncurry f)) :
+    Measurable fun y => ∫⁻ x, f x y ∂κ y :=
+  hf.lintegral_kernel_prod_left'
+#align measurable.lintegral_kernel_prod_left Measurable.lintegral_kernel_prod_left
+
+theorem Measurable.set_lintegral_kernel_prod_left {f : β → α → ℝ≥0∞} (hf : Measurable (uncurry f))
+    {s : Set β} (hs : MeasurableSet s) : Measurable fun b => ∫⁻ a in s, f a b ∂κ b :=
   by
   simp_rw [← lintegral_restrict κ hs]
-  exact measurable_lintegral _ hf
-#align probability_theory.kernel.measurable_set_lintegral ProbabilityTheory.kernel.measurableSet_lintegral
+  exact hf.lintegral_kernel_prod_left
+#align measurable.set_lintegral_kernel_prod_left Measurable.set_lintegral_kernel_prod_left
+
+theorem Measurable.lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) :
+    Measurable fun a => ∫⁻ b, f b ∂κ a :=
+  Measurable.lintegral_kernel_prod_right (hf.comp measurable_snd)
+#align measurable.lintegral_kernel Measurable.lintegral_kernel
 
-theorem measurableSet_lintegral' (κ : kernel α β) [IsSFiniteKernel κ] {f : β → ℝ≥0∞}
-    (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) :
-    Measurable fun a => ∫⁻ b in s, f b ∂κ a :=
-  measurableSet_lintegral κ (hf.comp measurable_snd) hs
-#align probability_theory.kernel.measurable_set_lintegral' ProbabilityTheory.kernel.measurableSet_lintegral'
+theorem Measurable.set_lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) {s : Set β}
+    (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f b ∂κ a :=
+  Measurable.set_lintegral_kernel_prod_right (hf.comp measurable_snd) hs
+#align measurable.set_lintegral_kernel Measurable.set_lintegral_kernel
+
+end Lintegral
+
+variable {E : Type _} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKernel η]
+
+theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
+    MeasurableSet { x | Integrable (f x) (κ x) } :=
+  by
+  simp_rw [integrable, hf.of_uncurry_left.ae_strongly_measurable, true_and_iff]
+  exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
+#align probability_theory.measurable_set_kernel_integrable ProbabilityTheory.measurableSet_kernel_integrable
+
+end ProbabilityTheory
+
+open ProbabilityTheory ProbabilityTheory.kernel
+
+namespace MeasureTheory
+
+variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [IsSFiniteKernel κ]
+  [IsSFiniteKernel η]
+
+theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
+    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
+  classical
+    borelize E
+    haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
+      hf.separable_space_range_union_singleton
+    let s : ℕ → simple_func (α × β) E :=
+      simple_func.approx_on _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
+    let s' : ℕ → α → simple_func β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
+    let f' : ℕ → α → E := fun n =>
+      { x | integrable (f x) (κ x) }.indicator fun x => (s' n x).integral (κ x)
+    have hf' : ∀ n, strongly_measurable (f' n) :=
+      by
+      intro n
+      refine' strongly_measurable.indicator _ (measurable_set_kernel_integrable hf)
+      have : ∀ x, ((s' n x).range.filterₓ fun x => x ≠ 0) ⊆ (s n).range :=
+        by
+        intro x
+        refine' Finset.Subset.trans (Finset.filter_subset _ _) _
+        intro y
+        simp_rw [simple_func.mem_range]
+        rintro ⟨z, rfl⟩
+        exact ⟨(x, z), rfl⟩
+      simp only [simple_func.integral_eq_sum_of_subset (this _)]
+      refine' Finset.stronglyMeasurable_sum _ fun x _ => _
+      refine' (Measurable.eNNReal_toReal _).StronglyMeasurable.smul_const _
+      simp (config := { singlePass := true }) only [simple_func.coe_comp, preimage_comp]
+      apply measurable_kernel_prod_mk_left
+      exact (s n).measurableSet_fiber x
+    have h2f' : tendsto f' at_top (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) :=
+      by
+      rw [tendsto_pi_nhds]
+      intro x
+      by_cases hfx : integrable (f x) (κ x)
+      · have : ∀ n, integrable (s' n x) (κ x) := by
+          intro n
+          apply (hfx.norm.add hfx.norm).mono' (s' n x).AeStronglyMeasurable
+          apply eventually_of_forall
+          intro y
+          simp_rw [s', simple_func.coe_comp]
+          exact simple_func.norm_approx_on_zero_le _ _ (x, y) n
+        simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
+          mem_set_of_eq]
+        refine'
+          tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
+            (fun n => (s' n x).AeStronglyMeasurable) (hfx.norm.add hfx.norm) _ _
+        ·
+          exact fun n =>
+            eventually_of_forall fun y => simple_func.norm_approx_on_zero_le _ _ (x, y) n
+        · refine' eventually_of_forall fun y => simple_func.tendsto_approx_on _ _ _
+          apply subset_closure
+          simp [-uncurry_apply_pair]
+      · simp [f', hfx, integral_undef]
+    exact stronglyMeasurable_of_tendsto _ hf' h2f'
+#align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
+
+theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
+    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x :=
+  by
+  rw [← uncurry_curry f] at hf
+  exact hf.integral_kernel_prod_right
+#align measure_theory.strongly_measurable.integral_kernel_prod_right' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
+
+theorem StronglyMeasurable.integral_kernel_prod_right'' {f : β × γ → E}
+    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂η (a, x) :=
+  by
+  change
+    strongly_measurable
+      ((fun x => ∫ y, (fun u : (α × β) × γ => f (u.1.2, u.2)) (x, y) ∂η x) ∘ fun x => (a, x))
+  refine' strongly_measurable.comp_measurable _ measurable_prod_mk_left
+  refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' _
+  exact hf.comp_measurable (measurable_fst.snd.prod_mk measurable_snd)
+#align measure_theory.strongly_measurable.integral_kernel_prod_right'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right''
+
+theorem StronglyMeasurable.integral_kernel_prod_left ⦃f : β → α → E⦄
+    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂κ y :=
+  (hf.comp_measurable measurable_swap).integral_kernel_prod_right'
+#align measure_theory.strongly_measurable.integral_kernel_prod_left MeasureTheory.StronglyMeasurable.integral_kernel_prod_left
+
+theorem StronglyMeasurable.integral_kernel_prod_left' ⦃f : β × α → E⦄ (hf : StronglyMeasurable f) :
+    StronglyMeasurable fun y => ∫ x, f (x, y) ∂κ y :=
+  (hf.comp_measurable measurable_swap).integral_kernel_prod_right'
+#align measure_theory.strongly_measurable.integral_kernel_prod_left' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'
+
+theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf : StronglyMeasurable f) :
+    StronglyMeasurable fun y => ∫ x, f (x, y) ∂η (a, y) :=
+  by
+  change
+    strongly_measurable
+      ((fun y => ∫ x, (fun u : γ × α × β => f (u.1, u.2.2)) (x, y) ∂η y) ∘ fun x => (a, x))
+  refine' strongly_measurable.comp_measurable _ measurable_prod_mk_left
+  refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' _
+  exact hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd)
+#align measure_theory.strongly_measurable.integral_kernel_prod_left'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
 
-end ProbabilityTheory.kernel
+end MeasureTheory
 

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

@@ -37,7 +37,7 @@ include mα mβ
 
 /-- This is an auxiliary lemma for `measurable_prod_mk_mem`. -/
 theorem measurable_prod_mk_mem_of_finite (κ : kernel α β) {t : Set (α × β)} (ht : MeasurableSet t)
-    (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a { b | (a, b) ∈ t } :=
+    (hκs : ∀ a, FiniteMeasure (κ a)) : Measurable fun a => κ a { b | (a, b) ∈ t } :=
   by
   -- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
   -- by boxes to prove the result by induction.

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

A PR accompanying #12339.

Zulip discussion

@@ -321,9 +321,9 @@ theorem StronglyMeasurable.integral_kernel_prod_right'' {f : β × γ → E}
   -- Porting note: was (`Function.comp` reducibility)
   -- refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' _
   -- exact hf.comp_measurable (measurable_fst.snd.prod_mk measurable_snd)
-  have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' (κ := η)
-    (hf.comp_measurable (measurable_fst.snd.prod_mk measurable_snd))
-  simpa using this
+  · have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' (κ := η)
+      (hf.comp_measurable (measurable_fst.snd.prod_mk measurable_snd))
+    simpa using this
 #align measure_theory.strongly_measurable.integral_kernel_prod_right'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_right''
 
 theorem StronglyMeasurable.integral_kernel_prod_left ⦃f : β → α → E⦄
@@ -345,9 +345,9 @@ theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf
   -- Porting note: was (`Function.comp` reducibility)
   -- refine' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' _
   -- exact hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd)
-  have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' (κ := η)
-    (hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd))
-  simpa using this
+  · have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' (κ := η)
+      (hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd))
+    simpa using this
 #align measure_theory.strongly_measurable.integral_kernel_prod_left'' MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
 
 end MeasureTheory

Let κ : kernel α (β × Ω) be a finite kernel, where Ω is a standard Borel space. Then if α is countable or β has a countably generated σ-algebra (for example if it is standard Borel), then there exists a kernel (α × β) Ω called conditional kernel and denoted by condKernel κ such that κ = fst κ ⊗ₖ condKernel κ.

Properties of integrals involving condKernel are collated in the file Integral.lean. The conditional kernel is unique (almost everywhere w.r.t. fst κ): this is proved in the file Unique.lean.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -253,12 +253,14 @@ open ProbabilityTheory ProbabilityTheory.kernel
 
 namespace MeasureTheory
 
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [IsSFiniteKernel κ]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [IsSFiniteKernel κ]
   [IsSFiniteKernel η]
 
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
   classical
+  by_cases hE : CompleteSpace E; swap
+  · simp [integral, hE, stronglyMeasurable_const]
   borelize E
   haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
     hf.separableSpace_range_union_singleton

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

@@ -234,7 +234,7 @@ theorem _root_.Measurable.set_lintegral_kernel {f : β → ℝ≥0∞} (hf : Mea
     (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f b ∂κ a := by
   -- Porting note: was term mode proof (`Function.comp` reducibility)
   refine Measurable.set_lintegral_kernel_prod_right ?_ hs
-  convert (hf.comp measurable_snd)
+  convert hf.comp measurable_snd
 #align measurable.set_lintegral_kernel Measurable.set_lintegral_kernel
 
 end Lintegral

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.

@@ -281,9 +281,9 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
   have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) := by
     rw [tendsto_pi_nhds]; intro x
     by_cases hfx : Integrable (f x) (κ x)
-    · have : ∀ n, Integrable (s' n x) (κ x) := by
-        intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
-        apply eventually_of_forall; intro y
+    · have (n) : Integrable (s' n x) (κ x) := by
+        apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
+        filter_upwards with y
         simp_rw [s', SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
       simp only [f',  hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
         mem_setOf_eq]

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

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

@@ -275,7 +275,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]
     refine' Finset.stronglyMeasurable_sum _ fun x _ => _
     refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _
-    simp only [SimpleFunc.coe_comp, preimage_comp]
+    simp only [s', SimpleFunc.coe_comp, preimage_comp]
     apply kernel.measurable_kernel_prod_mk_left
     exact (s n).measurableSet_fiber x
   have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) := by
@@ -284,8 +284,8 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     · have : ∀ n, Integrable (s' n x) (κ x) := by
         intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
         apply eventually_of_forall; intro y
-        simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
-      simp only [ hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
+        simp_rw [s', SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
+      simp only [f',  hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
         mem_setOf_eq]
       refine'
         tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
@@ -300,7 +300,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
         refine' eventually_of_forall fun y => SimpleFunc.tendsto_approxOn hf.measurable (by simp) _
         apply subset_closure
         simp [-uncurry_apply_pair]
-    · simp [hfx, integral_undef]
+    · simp [f', hfx, integral_undef]
   exact stronglyMeasurable_of_tendsto _ hf' h2f'
 #align measure_theory.strongly_measurable.integral_kernel_prod_right MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
 

Suggested by @loefflerd. Only code motion (and cosmetic adaptions, such as minimising import and open statements).

Pre-requisite for #11108 and (morally) #11110.

@@ -5,6 +5,7 @@ Authors: Rémy Degenne
 -/
 import Mathlib.Probability.Kernel.Basic
 import Mathlib.MeasureTheory.Constructions.Prod.Basic
+import Mathlib.MeasureTheory.Integral.DominatedConvergence
 
 #align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
 

Classifies by adding issue number (#10691) to porting notes claiming was rw.

@@ -175,7 +175,7 @@ theorem _root_.Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0
         (fun a => ∫⁻ b, g₁ (a, b) ∂κ a) + fun a => ∫⁻ b, g₂ (a, b) ∂κ a := by
       ext1 a
       rw [Pi.add_apply]
-      -- Porting note: was `rw` (`Function.comp` reducibility)
+      -- Porting note (#10691): was `rw` (`Function.comp` reducibility)
       erw [lintegral_add_left (g₁.measurable.comp measurable_prod_mk_left)]
       simp_rw [Function.comp_apply]
     rw [h_add]

Of the 18 uses of singlePass, in 3 cases we can just use rw, in 4 cases it isn't needed at all.

In the other 11 cases we are always use it as simp (config := {singlePass := true}) only [X] (i.e. with just a single simp lemma), and that simp call would loop forever (usually failing with a maxRecDepth error, sometimes with heartbeats). I've left these as is.

There's also one case where there was a missing only.

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

@@ -274,7 +274,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]
     refine' Finset.stronglyMeasurable_sum _ fun x _ => _
     refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _
-    simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]
+    simp only [SimpleFunc.coe_comp, preimage_comp]
     apply kernel.measurable_kernel_prod_mk_left
     exact (s n).measurableSet_fiber x
   have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) := by

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

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

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
 import Mathlib.Probability.Kernel.Basic
+import Mathlib.MeasureTheory.Constructions.Prod.Basic
 
 #align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
 

Removing from Mathlib.Init.Data.Prod from the early parts of the import hierarchy.

While at it, remove unnecessary uses of Prod.mk.eta across the library.

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

@@ -185,7 +185,7 @@ theorem _root_.Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0
     Measurable fun a => ∫⁻ b, f (a, b) ∂κ a := by
   refine' Measurable.lintegral_kernel_prod_right _
   have : (uncurry fun (a : α) (b : β) => f (a, b)) = f := by
-    ext x; rw [← @Prod.mk.eta _ _ x, uncurry_apply_pair]
+    ext x; rw [uncurry_apply_pair]
   rwa [this]
 #align measurable.lintegral_kernel_prod_right' Measurable.lintegral_kernel_prod_right'
 

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

@@ -133,8 +133,9 @@ variable [IsSFiniteKernel κ] [IsSFiniteKernel η]
 /-- Auxiliary lemma for `Measurable.lintegral_kernel_prod_right`. -/
 theorem kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
     (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a := by
-  simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
-  -- Porting note: `simp_rw` doesn't take, added the `conv` below
+  -- Porting note: was originally by
+  -- `simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]`
+  -- but this has no effect, so added the `conv` below
   conv =>
     congr
     ext

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

This has nice performance benefits.

@@ -29,7 +29,7 @@ open MeasureTheory ProbabilityTheory Function Set Filter
 
 open scoped MeasureTheory ENNReal Topology
 
-variable {α β γ : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
+variable {α β γ : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
   {κ : kernel α β} {η : kernel (α × β) γ} {a : α}
 
 namespace ProbabilityTheory
@@ -236,7 +236,7 @@ theorem _root_.Measurable.set_lintegral_kernel {f : β → ℝ≥0∞} (hf : Mea
 
 end Lintegral
 
-variable {E : Type _} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKernel η]
+variable {E : Type*} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
 theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
     MeasurableSet {x | Integrable (f x) (κ x)} := by
@@ -250,7 +250,7 @@ open ProbabilityTheory ProbabilityTheory.kernel
 
 namespace MeasureTheory
 
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [IsSFiniteKernel κ]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [IsSFiniteKernel κ]
   [IsSFiniteKernel η]
 
 theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.measurable_integral
-! leanprover-community/mathlib commit 28b2a92f2996d28e580450863c130955de0ed398
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Probability.Kernel.Basic
 
+#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
+
 /-!
 # Measurability of the integral against a kernel
 

@@ -151,7 +151,7 @@ measurable. -/
 theorem _root_.Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞}
     (hf : Measurable (uncurry f)) : Measurable fun a => ∫⁻ b, f a b ∂κ a := by
   let F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
-  have h : ∀ a, (⨆ n, F n a) = uncurry f a := SimpleFunc.iSup_eapprox_apply (uncurry f) hf
+  have h : ∀ a, ⨆ n, F n a = uncurry f a := SimpleFunc.iSup_eapprox_apply (uncurry f) hf
   simp only [Prod.forall, uncurry_apply_pair] at h
   simp_rw [← h]
   have : ∀ a, (∫⁻ b, ⨆ n, F n (a, b) ∂κ a) = ⨆ n, ∫⁻ b, F n (a, b) ∂κ a := by

Co-authored-by: int-y1 <jason_yuen2007@hotmail.com>

@@ -37,7 +37,7 @@ variable {α β γ : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace 
 
 namespace ProbabilityTheory
 
-namespace Kernel
+namespace kernel
 
 /-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
 theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
@@ -98,7 +98,7 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
       · exact fun i => (@measurable_prod_mk_left α β _ _ a) (hf_meas i)
     rw [h_tsum]
     exact Measurable.ennreal_tsum hf
-#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
+#align probability_theory.kernel.measurable_kernel_prod_mk_left_of_finite ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite
 
 theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
     (ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by
@@ -109,7 +109,7 @@ theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
   simp_rw [this]
   refine' Measurable.ennreal_tsum fun n => _
   exact measurable_kernel_prod_mk_left_of_finite ht inferInstance
-#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left
+#align probability_theory.kernel.measurable_kernel_prod_mk_left ProbabilityTheory.kernel.measurable_kernel_prod_mk_left
 
 theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
     (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) := by
@@ -118,14 +118,14 @@ theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)
   simp_rw [this]
   refine' (measurable_kernel_prod_mk_left _).comp measurable_prod_mk_left
   exact (measurable_fst.snd.prod_mk measurable_snd) hs
-#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'
+#align probability_theory.kernel.measurable_kernel_prod_mk_left' ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'
 
 theorem measurable_kernel_prod_mk_right [IsSFiniteKernel κ] {s : Set (β × α)}
     (hs : MeasurableSet s) : Measurable fun y => κ y ((fun x => (x, y)) ⁻¹' s) :=
   measurable_kernel_prod_mk_left (measurableSet_swap_iff.mpr hs)
-#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right
+#align probability_theory.kernel.measurable_kernel_prod_mk_right ProbabilityTheory.kernel.measurable_kernel_prod_mk_right
 
-end Kernel
+end kernel
 
 open ProbabilityTheory.kernel
 
@@ -134,7 +134,7 @@ section Lintegral
 variable [IsSFiniteKernel κ] [IsSFiniteKernel η]
 
 /-- Auxiliary lemma for `Measurable.lintegral_kernel_prod_right`. -/
-theorem Kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
+theorem kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
     (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a := by
   simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
   -- Porting note: `simp_rw` doesn't take, added the `conv` below
@@ -143,7 +143,7 @@ theorem Kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : M
     ext
     erw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
   exact Measurable.const_mul (measurable_kernel_prod_mk_left ht) c
-#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.Kernel.measurable_lintegral_indicator_const
+#align probability_theory.kernel.measurable_lintegral_indicator_const ProbabilityTheory.kernel.measurable_lintegral_indicator_const
 
 /-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is measurable when seen as a
 map from `α × β` (hypothesis `Measurable (uncurry f)`), the integral `a ↦ ∫⁻ b, f a b ∂(κ a)` is
@@ -168,7 +168,7 @@ theorem _root_.Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0
   · intro c t ht
     simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
       SimpleFunc.coe_zero, Set.piecewise_eq_indicator]
-    exact Kernel.measurable_lintegral_indicator_const (κ := κ) ht c
+    exact kernel.measurable_lintegral_indicator_const (κ := κ) ht c
   · intro g₁ g₂ _ hm₁ hm₂
     simp only [SimpleFunc.coe_add, Pi.add_apply]
     have h_add :
@@ -276,7 +276,7 @@ theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
     refine' Finset.stronglyMeasurable_sum _ fun x _ => _
     refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _
     simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]
-    apply Kernel.measurable_kernel_prod_mk_left
+    apply kernel.measurable_kernel_prod_mk_left
     exact (s n).measurableSet_fiber x
   have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) := by
     rw [tendsto_pi_nhds]; intro x

Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>