probability.kernel.basic ⟷ Mathlib.Probability.Kernel.Basic

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

@@ -520,23 +520,23 @@ theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
 -/
 
-#print ProbabilityTheory.kernel.set_integral_deterministic' /-
-theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+#print ProbabilityTheory.kernel.setIntegral_deterministic' /-
+theorem setIntegral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
-#align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
+#align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.setIntegral_deterministic'
 -/
 
-#print ProbabilityTheory.kernel.set_integral_deterministic /-
+#print ProbabilityTheory.kernel.setIntegral_deterministic /-
 @[simp]
-theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac f _ s]
-#align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
+#align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.setIntegral_deterministic
 -/
 
 end Deterministic
@@ -594,12 +594,12 @@ theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 -/
 
-#print ProbabilityTheory.kernel.set_integral_const /-
+#print ProbabilityTheory.kernel.setIntegral_const /-
 @[simp]
-theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem setIntegral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
     ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
-#align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
+#align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.setIntegral_const
 -/
 
 end Const
@@ -667,13 +667,13 @@ theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : 
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
 -/
 
-#print ProbabilityTheory.kernel.set_integral_restrict /-
+#print ProbabilityTheory.kernel.setIntegral_restrict /-
 @[simp]
-theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
+theorem setIntegral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
     ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, measure.restrict_restrict' hs]
-#align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
+#align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.setIntegral_restrict
 -/
 
 #print ProbabilityTheory.kernel.IsFiniteKernel.restrict /-
@@ -858,12 +858,12 @@ theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 -/
 
-#print ProbabilityTheory.kernel.set_integral_piecewise /-
-theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+#print ProbabilityTheory.kernel.setIntegral_piecewise /-
+theorem setIntegral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
     ∫ b in t, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
-#align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
+#align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.setIntegral_piecewise
 -/
 
 end Piecewise

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

@@ -239,7 +239,7 @@ theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b
     κ = η := by
   ext a s hs
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
-  simp_rw [lintegral_indicator_const hs, one_mul] at h 
+  simp_rw [lintegral_indicator_const hs, one_mul] at h
   rw [h]
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 -/

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

@@ -387,7 +387,15 @@ instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
 
 #print ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum /-
 theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
-    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by classical
+    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
+  classical
+  induction' I using Finset.induction with i I hi_nmem_I h_ind h
+  · rw [Finset.sum_empty]; infer_instance
+  · rw [Finset.sum_insert hi_nmem_I]
+    haveI : is_s_finite_kernel (κs i) := h i (Finset.mem_insert_self _ _)
+    have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
+      h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
+    exact is_s_finite_kernel.add _ _
 #align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum
 -/
 

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

@@ -387,15 +387,7 @@ instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
 
 #print ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum /-
 theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
-    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
-  classical
-  induction' I using Finset.induction with i I hi_nmem_I h_ind h
-  · rw [Finset.sum_empty]; infer_instance
-  · rw [Finset.sum_insert hi_nmem_I]
-    haveI : is_s_finite_kernel (κs i) := h i (Finset.mem_insert_self _ _)
-    have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
-      h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
-    exact is_s_finite_kernel.add _ _
+    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by classical
 #align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum
 -/
 

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathbin.MeasureTheory.Integral.Bochner
-import Mathbin.MeasureTheory.Constructions.Prod.Basic
+import MeasureTheory.Integral.Bochner
+import MeasureTheory.Constructions.Prod.Basic
 
 #align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 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.basic
-! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Integral.Bochner
 import Mathbin.MeasureTheory.Constructions.Prod.Basic
 
+#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
 /-!
 # Markov Kernels
 

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

@@ -240,7 +240,7 @@ theorem ext_iff' {η : kernel α β} :
 #print ProbabilityTheory.kernel.ext_fun /-
 theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
     κ = η := by
-  ext (a s hs)
+  ext a s hs
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
   simp_rw [lintegral_indicator_const hs, one_mul] at h 
   rw [h]
@@ -298,7 +298,7 @@ theorem sum_apply' [Countable ι] (κ : ι → kernel α β) (a : α) {s : Set 
 @[simp]
 theorem sum_zero [Countable ι] : (kernel.sum fun i : ι => (0 : kernel α β)) = 0 :=
   by
-  ext (a s hs) : 2
+  ext a s hs : 2
   rw [sum_apply' _ a hs]
   simp only [zero_apply, measure.coe_zero, Pi.zero_apply, tsum_zero]
 #align probability_theory.kernel.sum_zero ProbabilityTheory.kernel.sum_zero
@@ -307,13 +307,13 @@ theorem sum_zero [Countable ι] : (kernel.sum fun i : ι => (0 : kernel α β))
 #print ProbabilityTheory.kernel.sum_comm /-
 theorem sum_comm [Countable ι] (κ : ι → ι → kernel α β) :
     (kernel.sum fun n => kernel.sum (κ n)) = kernel.sum fun m => kernel.sum fun n => κ n m := by
-  ext (a s hs); simp_rw [sum_apply]; rw [measure.sum_comm]
+  ext a s hs; simp_rw [sum_apply]; rw [measure.sum_comm]
 #align probability_theory.kernel.sum_comm ProbabilityTheory.kernel.sum_comm
 -/
 
 #print ProbabilityTheory.kernel.sum_fintype /-
 @[simp]
-theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i := by ext (a s hs);
+theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i := by ext a s hs;
   simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
 #align probability_theory.kernel.sum_fintype ProbabilityTheory.kernel.sum_fintype
 -/
@@ -322,7 +322,7 @@ theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = 
 theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
     (kernel.sum fun n => κ n + η n) = kernel.sum κ + kernel.sum η :=
   by
-  ext (a s hs)
+  ext a s hs
   simp only [coe_fn_add, Pi.add_apply, sum_apply, measure.sum_apply _ hs, Pi.add_apply,
     measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
 #align probability_theory.kernel.sum_add ProbabilityTheory.kernel.sum_add
@@ -344,7 +344,7 @@ instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ
     IsSFiniteKernel κ :=
   ⟨⟨fun n => if n = 0 then κ else 0, fun n => by split_ifs; exact h; infer_instance,
       by
-      ext (a s hs)
+      ext a s hs
       rw [kernel.sum_apply' _ _ hs]
       have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by ext1 i;
         split_ifs <;> rfl
@@ -411,7 +411,7 @@ theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel
   refine' ⟨⟨fun n => seq (κs (e n).1) (e n).2, inferInstance, _⟩⟩
   have hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) := by
     simp_rw [kernel_sum_seq]
-  ext (a s hs) : 2
+  ext a s hs : 2
   rw [hκ_eq]
   simp_rw [kernel.sum_apply' _ _ hs]
   change ∑' (i) (m), seq (κs i) m a s = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)

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

@@ -78,29 +78,29 @@ instance {α β : Type _} [MeasurableSpace α] [MeasurableSpace β] :
 
 variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
 
-include mα mβ
-
 namespace Kernel
 
+#print ProbabilityTheory.kernel.coeFn_zero /-
 @[simp]
 theorem coeFn_zero : ⇑(0 : kernel α β) = 0 :=
   rfl
 #align probability_theory.kernel.coe_fn_zero ProbabilityTheory.kernel.coeFn_zero
+-/
 
+#print ProbabilityTheory.kernel.coeFn_add /-
 @[simp]
 theorem coeFn_add (κ η : kernel α β) : ⇑(κ + η) = κ + η :=
   rfl
 #align probability_theory.kernel.coe_fn_add ProbabilityTheory.kernel.coeFn_add
+-/
 
-omit mα mβ
-
+#print ProbabilityTheory.kernel.coeAddHom /-
 /-- Coercion to a function as an additive monoid homomorphism. -/
 def coeAddHom (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :
     kernel α β →+ α → Measure β :=
   ⟨coeFn, coeFn_zero, coeFn_add⟩
 #align probability_theory.kernel.coe_add_hom ProbabilityTheory.kernel.coeAddHom
-
-include mα mβ
+-/
 
 #print ProbabilityTheory.kernel.zero_apply /-
 @[simp]
@@ -109,51 +109,69 @@ theorem zero_apply (a : α) : (0 : kernel α β) a = 0 :=
 #align probability_theory.kernel.zero_apply ProbabilityTheory.kernel.zero_apply
 -/
 
+#print ProbabilityTheory.kernel.coe_finset_sum /-
 @[simp]
 theorem coe_finset_sum (I : Finset ι) (κ : ι → kernel α β) : ⇑(∑ i in I, κ i) = ∑ i in I, κ i :=
   (coeAddHom α β).map_sum _ _
 #align probability_theory.kernel.coe_finset_sum ProbabilityTheory.kernel.coe_finset_sum
+-/
 
+#print ProbabilityTheory.kernel.finset_sum_apply /-
 theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :
     (∑ i in I, κ i) a = ∑ i in I, κ i a := by rw [coe_finset_sum, Finset.sum_apply]
 #align probability_theory.kernel.finset_sum_apply ProbabilityTheory.kernel.finset_sum_apply
+-/
 
+#print ProbabilityTheory.kernel.finset_sum_apply' /-
 theorem finset_sum_apply' (I : Finset ι) (κ : ι → kernel α β) (a : α) (s : Set β) :
     (∑ i in I, κ i) a s = ∑ i in I, κ i a s := by rw [finset_sum_apply, measure.finset_sum_apply]
 #align probability_theory.kernel.finset_sum_apply' ProbabilityTheory.kernel.finset_sum_apply'
+-/
 
 end Kernel
 
+#print ProbabilityTheory.IsMarkovKernel /-
 /-- A kernel is a Markov kernel if every measure in its image is a probability measure. -/
 class IsMarkovKernel (κ : kernel α β) : Prop where
   IsProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
 #align probability_theory.is_markov_kernel ProbabilityTheory.IsMarkovKernel
+-/
 
+#print ProbabilityTheory.IsFiniteKernel /-
 /-- A kernel is finite if every measure in its image is finite, with a uniform bound. -/
 class IsFiniteKernel (κ : kernel α β) : Prop where
   exists_univ_le : ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ a, κ a Set.univ ≤ C
 #align probability_theory.is_finite_kernel ProbabilityTheory.IsFiniteKernel
+-/
 
+#print ProbabilityTheory.IsFiniteKernel.bound /-
 /-- A constant `C : ℝ≥0∞` such that `C < ∞` (`is_finite_kernel.bound_lt_top κ`) and for all
 `a : α` and `s : set β`, `κ a s ≤ C` (`measure_le_bound κ a s`). -/
 noncomputable def IsFiniteKernel.bound (κ : kernel α β) [h : IsFiniteKernel κ] : ℝ≥0∞ :=
   h.exists_univ_le.some
 #align probability_theory.is_finite_kernel.bound ProbabilityTheory.IsFiniteKernel.bound
+-/
 
+#print ProbabilityTheory.IsFiniteKernel.bound_lt_top /-
 theorem IsFiniteKernel.bound_lt_top (κ : kernel α β) [h : IsFiniteKernel κ] :
     IsFiniteKernel.bound κ < ∞ :=
   h.exists_univ_le.choose_spec.1
 #align probability_theory.is_finite_kernel.bound_lt_top ProbabilityTheory.IsFiniteKernel.bound_lt_top
+-/
 
+#print ProbabilityTheory.IsFiniteKernel.bound_ne_top /-
 theorem IsFiniteKernel.bound_ne_top (κ : kernel α β) [h : IsFiniteKernel κ] :
     IsFiniteKernel.bound κ ≠ ∞ :=
   (IsFiniteKernel.bound_lt_top κ).Ne
 #align probability_theory.is_finite_kernel.bound_ne_top ProbabilityTheory.IsFiniteKernel.bound_ne_top
+-/
 
+#print ProbabilityTheory.kernel.measure_le_bound /-
 theorem kernel.measure_le_bound (κ : kernel α β) [h : IsFiniteKernel κ] (a : α) (s : Set β) :
     κ a s ≤ IsFiniteKernel.bound κ :=
   (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
 #align probability_theory.kernel.measure_le_bound ProbabilityTheory.kernel.measure_le_bound
+-/
 
 #print ProbabilityTheory.isFiniteKernel_zero /-
 instance isFiniteKernel_zero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
@@ -163,6 +181,7 @@ instance isFiniteKernel_zero (α β : Type _) {mα : MeasurableSpace α} {mβ :
 #align probability_theory.is_finite_kernel_zero ProbabilityTheory.isFiniteKernel_zero
 -/
 
+#print ProbabilityTheory.IsFiniteKernel.add /-
 instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :
     IsFiniteKernel (κ + η) :=
   by
@@ -173,38 +192,52 @@ instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFinite
   simp_rw [kernel.coe_fn_add, Pi.add_apply, measure.coe_add, Pi.add_apply]
   exact add_le_add (kernel.measure_le_bound _ _ _) (kernel.measure_le_bound _ _ _)
 #align probability_theory.is_finite_kernel.add ProbabilityTheory.IsFiniteKernel.add
+-/
 
 variable {κ : kernel α β}
 
+#print ProbabilityTheory.IsMarkovKernel.is_probability_measure' /-
 instance IsMarkovKernel.is_probability_measure' [h : IsMarkovKernel κ] (a : α) :
     IsProbabilityMeasure (κ a) :=
   IsMarkovKernel.isProbabilityMeasure a
 #align probability_theory.is_markov_kernel.is_probability_measure' ProbabilityTheory.IsMarkovKernel.is_probability_measure'
+-/
 
+#print ProbabilityTheory.IsFiniteKernel.isFiniteMeasure /-
 instance IsFiniteKernel.isFiniteMeasure [h : IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
   ⟨(kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
 #align probability_theory.is_finite_kernel.is_finite_measure ProbabilityTheory.IsFiniteKernel.isFiniteMeasure
+-/
 
+#print ProbabilityTheory.IsMarkovKernel.isFiniteKernel /-
 instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ] :
     IsFiniteKernel κ :=
   ⟨⟨1, ENNReal.one_lt_top, fun a => prob_le_one⟩⟩
 #align probability_theory.is_markov_kernel.is_finite_kernel ProbabilityTheory.IsMarkovKernel.isFiniteKernel
+-/
 
 namespace Kernel
 
+#print ProbabilityTheory.kernel.ext /-
 @[ext]
 theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η := by ext1; ext1 a; exact h a
 #align probability_theory.kernel.ext ProbabilityTheory.kernel.ext
+-/
 
+#print ProbabilityTheory.kernel.ext_iff /-
 theorem ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a :=
   ⟨fun h a => by rw [h], ext⟩
 #align probability_theory.kernel.ext_iff ProbabilityTheory.kernel.ext_iff
+-/
 
+#print ProbabilityTheory.kernel.ext_iff' /-
 theorem ext_iff' {η : kernel α β} :
     κ = η ↔ ∀ (a) (s : Set β) (hs : MeasurableSet s), κ a s = η a s := by
   simp_rw [ext_iff, measure.ext_iff]
 #align probability_theory.kernel.ext_iff' ProbabilityTheory.kernel.ext_iff'
+-/
 
+#print ProbabilityTheory.kernel.ext_fun /-
 theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
     κ = η := by
   ext (a s hs)
@@ -212,23 +245,31 @@ theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b
   simp_rw [lintegral_indicator_const hs, one_mul] at h 
   rw [h]
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
+-/
 
+#print ProbabilityTheory.kernel.ext_fun_iff /-
 theorem ext_fun_iff {η : kernel α β} :
     κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a :=
   ⟨fun h a f hf => by rw [h], ext_fun⟩
 #align probability_theory.kernel.ext_fun_iff ProbabilityTheory.kernel.ext_fun_iff
+-/
 
+#print ProbabilityTheory.kernel.measurable /-
 protected theorem measurable (κ : kernel α β) : Measurable κ :=
   κ.Prop
 #align probability_theory.kernel.measurable ProbabilityTheory.kernel.measurable
+-/
 
+#print ProbabilityTheory.kernel.measurable_coe /-
 protected theorem measurable_coe (κ : kernel α β) {s : Set β} (hs : MeasurableSet s) :
     Measurable fun a => κ a s :=
   (Measure.measurable_coe hs).comp (kernel.measurable κ)
 #align probability_theory.kernel.measurable_coe ProbabilityTheory.kernel.measurable_coe
+-/
 
 section Sum
 
+#print ProbabilityTheory.kernel.sum /-
 /-- Sum of an indexed family of kernels. -/
 protected noncomputable def sum [Countable ι] (κ : ι → kernel α β) : kernel α β
     where
@@ -238,16 +279,22 @@ protected noncomputable def sum [Countable ι] (κ : ι → kernel α β) : kern
     simp_rw [measure.sum_apply _ hs]
     exact Measurable.ennreal_tsum fun n => kernel.measurable_coe (κ n) hs
 #align probability_theory.kernel.sum ProbabilityTheory.kernel.sum
+-/
 
+#print ProbabilityTheory.kernel.sum_apply /-
 theorem sum_apply [Countable ι] (κ : ι → kernel α β) (a : α) :
     kernel.sum κ a = Measure.sum fun n => κ n a :=
   rfl
 #align probability_theory.kernel.sum_apply ProbabilityTheory.kernel.sum_apply
+-/
 
+#print ProbabilityTheory.kernel.sum_apply' /-
 theorem sum_apply' [Countable ι] (κ : ι → kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s) :
     kernel.sum κ a s = ∑' n, κ n a s := by rw [sum_apply κ a, measure.sum_apply _ hs]
 #align probability_theory.kernel.sum_apply' ProbabilityTheory.kernel.sum_apply'
+-/
 
+#print ProbabilityTheory.kernel.sum_zero /-
 @[simp]
 theorem sum_zero [Countable ι] : (kernel.sum fun i : ι => (0 : kernel α β)) = 0 :=
   by
@@ -255,17 +302,23 @@ theorem sum_zero [Countable ι] : (kernel.sum fun i : ι => (0 : kernel α β))
   rw [sum_apply' _ a hs]
   simp only [zero_apply, measure.coe_zero, Pi.zero_apply, tsum_zero]
 #align probability_theory.kernel.sum_zero ProbabilityTheory.kernel.sum_zero
+-/
 
+#print ProbabilityTheory.kernel.sum_comm /-
 theorem sum_comm [Countable ι] (κ : ι → ι → kernel α β) :
     (kernel.sum fun n => kernel.sum (κ n)) = kernel.sum fun m => kernel.sum fun n => κ n m := by
   ext (a s hs); simp_rw [sum_apply]; rw [measure.sum_comm]
 #align probability_theory.kernel.sum_comm ProbabilityTheory.kernel.sum_comm
+-/
 
+#print ProbabilityTheory.kernel.sum_fintype /-
 @[simp]
 theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i := by ext (a s hs);
   simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
 #align probability_theory.kernel.sum_fintype ProbabilityTheory.kernel.sum_fintype
+-/
 
+#print ProbabilityTheory.kernel.sum_add /-
 theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
     (kernel.sum fun n => κ n + η n) = kernel.sum κ + kernel.sum η :=
   by
@@ -273,16 +326,20 @@ theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
   simp only [coe_fn_add, Pi.add_apply, sum_apply, measure.sum_apply _ hs, Pi.add_apply,
     measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
 #align probability_theory.kernel.sum_add ProbabilityTheory.kernel.sum_add
+-/
 
 end Sum
 
 section SFinite
 
+#print ProbabilityTheory.IsSFiniteKernel /-
 /-- A kernel is s-finite if it can be written as the sum of countably many finite kernels. -/
 class ProbabilityTheory.IsSFiniteKernel (κ : kernel α β) : Prop where
   tsum_finite : ∃ κs : ℕ → kernel α β, (∀ n, IsFiniteKernel (κs n)) ∧ κ = kernel.sum κs
 #align probability_theory.is_s_finite_kernel ProbabilityTheory.IsSFiniteKernel
+-/
 
+#print ProbabilityTheory.kernel.IsFiniteKernel.isSFiniteKernel /-
 instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
     IsSFiniteKernel κ :=
   ⟨⟨fun n => if n = 0 then κ else 0, fun n => by split_ifs; exact h; infer_instance,
@@ -293,33 +350,45 @@ instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ
         split_ifs <;> rfl
       rw [this, tsum_ite_eq]⟩⟩
 #align probability_theory.kernel.is_finite_kernel.is_s_finite_kernel ProbabilityTheory.kernel.IsFiniteKernel.isSFiniteKernel
+-/
 
+#print ProbabilityTheory.kernel.seq /-
 /-- A sequence of finite kernels such that `κ = kernel.sum (seq κ)`. See `is_finite_kernel_seq`
 and `kernel_sum_seq`. -/
 noncomputable def seq (κ : kernel α β) [h : IsSFiniteKernel κ] : ℕ → kernel α β :=
   h.tsum_finite.some
 #align probability_theory.kernel.seq ProbabilityTheory.kernel.seq
+-/
 
+#print ProbabilityTheory.kernel.kernel_sum_seq /-
 theorem kernel_sum_seq (κ : kernel α β) [h : IsSFiniteKernel κ] : kernel.sum (seq κ) = κ :=
   h.tsum_finite.choose_spec.2.symm
 #align probability_theory.kernel.kernel_sum_seq ProbabilityTheory.kernel.kernel_sum_seq
+-/
 
+#print ProbabilityTheory.kernel.measure_sum_seq /-
 theorem measure_sum_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (a : α) :
     (Measure.sum fun n => seq κ n a) = κ a := by rw [← kernel.sum_apply, kernel_sum_seq κ]
 #align probability_theory.kernel.measure_sum_seq ProbabilityTheory.kernel.measure_sum_seq
+-/
 
+#print ProbabilityTheory.kernel.isFiniteKernel_seq /-
 instance isFiniteKernel_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :
     IsFiniteKernel (kernel.seq κ n) :=
   h.tsum_finite.choose_spec.1 n
 #align probability_theory.kernel.is_finite_kernel_seq ProbabilityTheory.kernel.isFiniteKernel_seq
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.add /-
 instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
     IsSFiniteKernel (κ + η) :=
   by
   refine' ⟨⟨fun n => seq κ n + seq η n, fun n => inferInstance, _⟩⟩
   rw [sum_add, kernel_sum_seq κ, kernel_sum_seq η]
 #align probability_theory.kernel.is_s_finite_kernel.add ProbabilityTheory.kernel.IsSFiniteKernel.add
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum /-
 theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
     (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
   classical
@@ -331,8 +400,10 @@ theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
       h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
     exact is_s_finite_kernel.add _ _
 #align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i m) -/
+#print ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable /-
 theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel α β}
     (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (kernel.sum κs) :=
   by
@@ -348,7 +419,9 @@ theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel
   · rw [tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
   · infer_instance
 #align probability_theory.kernel.is_s_finite_kernel_sum_of_denumerable ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable
+-/
 
+#print ProbabilityTheory.kernel.isSFiniteKernel_sum /-
 theorem isSFiniteKernel_sum [Countable ι] {κs : ι → kernel α β}
     (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (kernel.sum κs) :=
   by
@@ -359,11 +432,13 @@ theorem isSFiniteKernel_sum [Countable ι] {κs : ι → kernel α β}
   haveI : Denumerable ι := Denumerable.ofEncodableOfInfinite ι
   exact is_s_finite_kernel_sum_of_denumerable hκs
 #align probability_theory.kernel.is_s_finite_kernel_sum ProbabilityTheory.kernel.isSFiniteKernel_sum
+-/
 
 end SFinite
 
 section Deterministic
 
+#print ProbabilityTheory.kernel.deterministic /-
 /-- Kernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel. -/
 noncomputable def deterministic (f : α → β) (hf : Measurable f) : kernel α β
     where
@@ -373,12 +448,16 @@ noncomputable def deterministic (f : α → β) (hf : Measurable f) : kernel α
     simp_rw [measure.dirac_apply' _ hs]
     exact measurable_one.indicator (hf hs)
 #align probability_theory.kernel.deterministic ProbabilityTheory.kernel.deterministic
+-/
 
+#print ProbabilityTheory.kernel.deterministic_apply /-
 theorem deterministic_apply {f : α → β} (hf : Measurable f) (a : α) :
     deterministic f hf a = Measure.dirac (f a) :=
   rfl
 #align probability_theory.kernel.deterministic_apply ProbabilityTheory.kernel.deterministic_apply
+-/
 
+#print ProbabilityTheory.kernel.deterministic_apply' /-
 theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : Set β}
     (hs : MeasurableSet s) : deterministic f hf a s = s.indicator (fun _ => 1) (f a) :=
   by
@@ -386,6 +465,7 @@ theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : S
   change measure.dirac (f a) s = s.indicator 1 (f a)
   simp_rw [measure.dirac_apply' _ hs]
 #align probability_theory.kernel.deterministic_apply' ProbabilityTheory.kernel.deterministic_apply'
+-/
 
 #print ProbabilityTheory.kernel.isMarkovKernel_deterministic /-
 instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
@@ -394,50 +474,65 @@ instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
 #align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
 -/
 
+#print ProbabilityTheory.kernel.lintegral_deterministic' /-
 theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     (hf : Measurable f) : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac' _ hf]
 #align probability_theory.kernel.lintegral_deterministic' ProbabilityTheory.kernel.lintegral_deterministic'
+-/
 
+#print ProbabilityTheory.kernel.lintegral_deterministic /-
 @[simp]
 theorem lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac (g a) f]
 #align probability_theory.kernel.lintegral_deterministic ProbabilityTheory.kernel.lintegral_deterministic
+-/
 
+#print ProbabilityTheory.kernel.set_lintegral_deterministic' /-
 theorem set_lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
     ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]
 #align probability_theory.kernel.set_lintegral_deterministic' ProbabilityTheory.kernel.set_lintegral_deterministic'
+-/
 
+#print ProbabilityTheory.kernel.set_lintegral_deterministic /-
 @[simp]
 theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
     ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac f s]
 #align probability_theory.kernel.set_lintegral_deterministic ProbabilityTheory.kernel.set_lintegral_deterministic
+-/
 
+#print ProbabilityTheory.kernel.integral_deterministic' /-
 theorem integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac' _ _ hf]
 #align probability_theory.kernel.integral_deterministic' ProbabilityTheory.kernel.integral_deterministic'
+-/
 
+#print ProbabilityTheory.kernel.integral_deterministic /-
 @[simp]
 theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac _ (g a)]
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
+-/
 
+#print ProbabilityTheory.kernel.set_integral_deterministic' /-
 theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
 #align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
+-/
 
+#print ProbabilityTheory.kernel.set_integral_deterministic /-
 @[simp]
 theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
@@ -445,21 +540,20 @@ theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSp
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac f _ s]
 #align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
+-/
 
 end Deterministic
 
 section Const
 
-omit mα mβ
-
+#print ProbabilityTheory.kernel.const /-
 /-- Constant kernel, which always returns the same measure. -/
 def const (α : Type _) {β : Type _} [MeasurableSpace α] {mβ : MeasurableSpace β} (μβ : Measure β) :
     kernel α β where
   val _ := μβ
   property := Measure.measurable_of_measurable_coe _ fun s hs => measurable_const
 #align probability_theory.kernel.const ProbabilityTheory.kernel.const
-
-include mα mβ
+-/
 
 #print ProbabilityTheory.kernel.const_apply /-
 theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
@@ -513,8 +607,7 @@ theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ
 
 end Const
 
-omit mα
-
+#print ProbabilityTheory.kernel.ofFunOfCountable /-
 /-- In a countable space with measurable singletons, every function `α → measure β` defines a
 kernel. -/
 def ofFunOfCountable [MeasurableSpace α] {mβ : MeasurableSpace β} [Countable α]
@@ -523,13 +616,13 @@ def ofFunOfCountable [MeasurableSpace α] {mβ : MeasurableSpace β} [Countable
   val := f
   property := measurable_of_countable f
 #align probability_theory.kernel.of_fun_of_countable ProbabilityTheory.kernel.ofFunOfCountable
-
-include mα
+-/
 
 section Restrict
 
 variable {s t : Set β}
 
+#print ProbabilityTheory.kernel.restrict /-
 /-- Kernel given by the restriction of the measures in the image of a kernel to a set. -/
 protected noncomputable def restrict (κ : kernel α β) (hs : MeasurableSet s) : kernel α β
     where
@@ -539,40 +632,54 @@ protected noncomputable def restrict (κ : kernel α β) (hs : MeasurableSet s)
     simp_rw [measure.restrict_apply ht]
     exact kernel.measurable_coe κ (ht.inter hs)
 #align probability_theory.kernel.restrict ProbabilityTheory.kernel.restrict
+-/
 
+#print ProbabilityTheory.kernel.restrict_apply /-
 theorem restrict_apply (κ : kernel α β) (hs : MeasurableSet s) (a : α) :
     kernel.restrict κ hs a = (κ a).restrict s :=
   rfl
 #align probability_theory.kernel.restrict_apply ProbabilityTheory.kernel.restrict_apply
+-/
 
+#print ProbabilityTheory.kernel.restrict_apply' /-
 theorem restrict_apply' (κ : kernel α β) (hs : MeasurableSet s) (a : α) (ht : MeasurableSet t) :
     kernel.restrict κ hs a t = (κ a) (t ∩ s) := by
   rw [restrict_apply κ hs a, measure.restrict_apply ht]
 #align probability_theory.kernel.restrict_apply' ProbabilityTheory.kernel.restrict_apply'
+-/
 
+#print ProbabilityTheory.kernel.restrict_univ /-
 @[simp]
 theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ := by ext1 a;
   rw [kernel.restrict_apply, measure.restrict_univ]
 #align probability_theory.kernel.restrict_univ ProbabilityTheory.kernel.restrict_univ
+-/
 
+#print ProbabilityTheory.kernel.lintegral_restrict /-
 @[simp]
 theorem lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞) :
     ∫⁻ b, f b ∂kernel.restrict κ hs a = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
 #align probability_theory.kernel.lintegral_restrict ProbabilityTheory.kernel.lintegral_restrict
+-/
 
+#print ProbabilityTheory.kernel.set_lintegral_restrict /-
 @[simp]
 theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞)
     (t : Set β) : ∫⁻ b in t, f b ∂kernel.restrict κ hs a = ∫⁻ b in t ∩ s, f b ∂κ a := by
   rw [restrict_apply, measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
+-/
 
+#print ProbabilityTheory.kernel.set_integral_restrict /-
 @[simp]
 theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
     ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
+-/
 
+#print ProbabilityTheory.kernel.IsFiniteKernel.restrict /-
 instance IsFiniteKernel.restrict (κ : kernel α β) [IsFiniteKernel κ] (hs : MeasurableSet s) :
     IsFiniteKernel (kernel.restrict κ hs) :=
   by
@@ -580,7 +687,9 @@ instance IsFiniteKernel.restrict (κ : kernel α β) [IsFiniteKernel κ] (hs : M
   rw [restrict_apply' κ hs a MeasurableSet.univ]
   exact measure_le_bound κ a _
 #align probability_theory.kernel.is_finite_kernel.restrict ProbabilityTheory.kernel.IsFiniteKernel.restrict
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.restrict /-
 instance IsSFiniteKernel.restrict (κ : kernel α β) [IsSFiniteKernel κ] (hs : MeasurableSet s) :
     IsSFiniteKernel (kernel.restrict κ hs) :=
   by
@@ -588,6 +697,7 @@ instance IsSFiniteKernel.restrict (κ : kernel α β) [IsSFiniteKernel κ] (hs :
   ext1 a
   simp_rw [sum_apply, restrict_apply, ← measure.restrict_sum _ hs, ← sum_apply, kernel_sum_seq]
 #align probability_theory.kernel.is_s_finite_kernel.restrict ProbabilityTheory.kernel.IsSFiniteKernel.restrict
+-/
 
 end Restrict
 
@@ -595,8 +705,7 @@ section ComapRight
 
 variable {γ : Type _} {mγ : MeasurableSpace γ} {f : γ → β}
 
-include mγ
-
+#print ProbabilityTheory.kernel.comapRight /-
 /-- Kernel with value `(κ a).comap f`, for a measurable embedding `f`. That is, for a measurable set
 `t : set β`, `comap_right κ hf a t = κ a (f '' t)`. -/
 noncomputable def comapRight (κ : kernel α β) (hf : MeasurableEmbedding f) : kernel α γ
@@ -612,18 +721,24 @@ noncomputable def comapRight (κ : kernel α β) (hf : MeasurableEmbedding f) :
     rw [this]
     exact kernel.measurable_coe _ (hf.measurable_set_image.mpr ht)
 #align probability_theory.kernel.comap_right ProbabilityTheory.kernel.comapRight
+-/
 
+#print ProbabilityTheory.kernel.comapRight_apply /-
 theorem comapRight_apply (κ : kernel α β) (hf : MeasurableEmbedding f) (a : α) :
     comapRight κ hf a = Measure.comap f (κ a) :=
   rfl
 #align probability_theory.kernel.comap_right_apply ProbabilityTheory.kernel.comapRight_apply
+-/
 
+#print ProbabilityTheory.kernel.comapRight_apply' /-
 theorem comapRight_apply' (κ : kernel α β) (hf : MeasurableEmbedding f) (a : α) {t : Set γ}
     (ht : MeasurableSet t) : comapRight κ hf a t = κ a (f '' t) := by
   rw [comap_right_apply,
     measure.comap_apply _ hf.injective (fun s => hf.measurable_set_image.mpr) _ ht]
 #align probability_theory.kernel.comap_right_apply' ProbabilityTheory.kernel.comapRight_apply'
+-/
 
+#print ProbabilityTheory.kernel.IsMarkovKernel.comapRight /-
 theorem IsMarkovKernel.comapRight (κ : kernel α β) (hf : MeasurableEmbedding f)
     (hκ : ∀ a, κ a (Set.range f) = 1) : IsMarkovKernel (comapRight κ hf) :=
   by
@@ -632,7 +747,9 @@ theorem IsMarkovKernel.comapRight (κ : kernel α β) (hf : MeasurableEmbedding
   simp only [Set.image_univ, Subtype.range_coe_subtype, Set.setOf_mem_eq]
   exact hκ a
 #align probability_theory.kernel.is_markov_kernel.comap_right ProbabilityTheory.kernel.IsMarkovKernel.comapRight
+-/
 
+#print ProbabilityTheory.kernel.IsFiniteKernel.comapRight /-
 instance IsFiniteKernel.comapRight (κ : kernel α β) [IsFiniteKernel κ]
     (hf : MeasurableEmbedding f) : IsFiniteKernel (comapRight κ hf) :=
   by
@@ -640,7 +757,9 @@ instance IsFiniteKernel.comapRight (κ : kernel α β) [IsFiniteKernel κ]
   rw [comap_right_apply' κ hf a MeasurableSet.univ]
   exact measure_le_bound κ a _
 #align probability_theory.kernel.is_finite_kernel.comap_right ProbabilityTheory.kernel.IsFiniteKernel.comapRight
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.comapRight /-
 instance IsSFiniteKernel.comapRight (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : MeasurableEmbedding f) : IsSFiniteKernel (comapRight κ hf) :=
   by
@@ -659,6 +778,7 @@ instance IsSFiniteKernel.comapRight (κ : kernel α β) [IsSFiniteKernel κ]
     rw [measure.comap_apply _ hf.injective (fun s' => hf.measurable_set_image.mpr) _ ht]
   rw [this, measure_sum_seq]
 #align probability_theory.kernel.is_s_finite_kernel.comap_right ProbabilityTheory.kernel.IsSFiniteKernel.comapRight
+-/
 
 end ComapRight
 
@@ -666,6 +786,7 @@ section Piecewise
 
 variable {η : kernel α β} {s : Set α} {hs : MeasurableSet s} [DecidablePred (· ∈ s)]
 
+#print ProbabilityTheory.kernel.piecewise /-
 /-- `piecewise hs κ η` is the kernel equal to `κ` on the measurable set `s` and to `η` on its
 complement. -/
 def piecewise (hs : MeasurableSet s) (κ η : kernel α β) : kernel α β
@@ -673,21 +794,29 @@ def piecewise (hs : MeasurableSet s) (κ η : kernel α β) : kernel α β
   val a := if a ∈ s then κ a else η a
   property := Measurable.piecewise hs (kernel.measurable _) (kernel.measurable _)
 #align probability_theory.kernel.piecewise ProbabilityTheory.kernel.piecewise
+-/
 
+#print ProbabilityTheory.kernel.piecewise_apply /-
 theorem piecewise_apply (a : α) : piecewise hs κ η a = if a ∈ s then κ a else η a :=
   rfl
 #align probability_theory.kernel.piecewise_apply ProbabilityTheory.kernel.piecewise_apply
+-/
 
+#print ProbabilityTheory.kernel.piecewise_apply' /-
 theorem piecewise_apply' (a : α) (t : Set β) :
     piecewise hs κ η a t = if a ∈ s then κ a t else η a t := by rw [piecewise_apply];
   split_ifs <;> rfl
 #align probability_theory.kernel.piecewise_apply' ProbabilityTheory.kernel.piecewise_apply'
+-/
 
+#print ProbabilityTheory.kernel.IsMarkovKernel.piecewise /-
 instance IsMarkovKernel.piecewise [IsMarkovKernel κ] [IsMarkovKernel η] :
     IsMarkovKernel (piecewise hs κ η) := by refine' ⟨fun a => ⟨_⟩⟩;
   rw [piecewise_apply', measure_univ, measure_univ, if_t_t]
 #align probability_theory.kernel.is_markov_kernel.piecewise ProbabilityTheory.kernel.IsMarkovKernel.piecewise
+-/
 
+#print ProbabilityTheory.kernel.IsFiniteKernel.piecewise /-
 instance IsFiniteKernel.piecewise [IsFiniteKernel κ] [IsFiniteKernel η] :
     IsFiniteKernel (piecewise hs κ η) :=
   by
@@ -696,7 +825,9 @@ instance IsFiniteKernel.piecewise [IsFiniteKernel κ] [IsFiniteKernel η] :
   rw [piecewise_apply']
   exact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _))
 #align probability_theory.kernel.is_finite_kernel.piecewise ProbabilityTheory.kernel.IsFiniteKernel.piecewise
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.piecewise /-
 instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKernel η] :
     IsSFiniteKernel (piecewise hs κ η) :=
   by
@@ -705,29 +836,38 @@ instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKernel η] :
   simp_rw [sum_apply, kernel.piecewise_apply]
   split_ifs <;> exact (measure_sum_seq _ a).symm
 #align probability_theory.kernel.is_s_finite_kernel.piecewise ProbabilityTheory.kernel.IsSFiniteKernel.piecewise
+-/
 
+#print ProbabilityTheory.kernel.lintegral_piecewise /-
 theorem lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
     ∫⁻ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.lintegral_piecewise ProbabilityTheory.kernel.lintegral_piecewise
+-/
 
+#print ProbabilityTheory.kernel.set_lintegral_piecewise /-
 theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
     ∫⁻ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫⁻ b in t, g b ∂κ a else ∫⁻ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
+-/
 
+#print ProbabilityTheory.kernel.integral_piecewise /-
 theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     (a : α) (g : β → E) :
     ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
+-/
 
+#print ProbabilityTheory.kernel.set_integral_piecewise /-
 theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
     ∫ b in t, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
+-/
 
 end Piecewise
 

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: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.basic
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.MeasureTheory.Constructions.Prod.Basic
 /-!
 # Markov Kernels
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A kernel from a measurable space `α` to another measurable space `β` is a measurable map
 `α → measure β`, where the measurable space instance on `measure β` is the one defined in
 `measure_theory.measure.measurable_space`. That is, a kernel `κ` verifies that for all measurable

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

@@ -202,7 +202,7 @@ theorem ext_iff' {η : kernel α β} :
   simp_rw [ext_iff, measure.ext_iff]
 #align probability_theory.kernel.ext_iff' ProbabilityTheory.kernel.ext_iff'
 
-theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a) :
+theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
     κ = η := by
   ext (a s hs)
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
@@ -211,7 +211,7 @@ theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 
 theorem ext_fun_iff {η : kernel α β} :
-    κ = η ↔ ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a :=
+    κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a :=
   ⟨fun h a f hf => by rw [h], ext_fun⟩
 #align probability_theory.kernel.ext_fun_iff ProbabilityTheory.kernel.ext_fun_iff
 
@@ -340,7 +340,7 @@ theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel
   ext (a s hs) : 2
   rw [hκ_eq]
   simp_rw [kernel.sum_apply' _ _ hs]
-  change (∑' (i) (m), seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
+  change ∑' (i) (m), seq (κs i) m a s = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
   rw [e.tsum_eq]
   · rw [tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
   · infer_instance
@@ -392,46 +392,46 @@ instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
 -/
 
 theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : Measurable f) : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    (hf : Measurable f) : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac' _ hf]
 #align probability_theory.kernel.lintegral_deterministic' ProbabilityTheory.kernel.lintegral_deterministic'
 
 @[simp]
 theorem lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    [MeasurableSingletonClass β] : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac (g a) f]
 #align probability_theory.kernel.lintegral_deterministic ProbabilityTheory.kernel.lintegral_deterministic
 
 theorem set_lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
-    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]
 #align probability_theory.kernel.set_lintegral_deterministic' ProbabilityTheory.kernel.set_lintegral_deterministic'
 
 @[simp]
 theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
-    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac f s]
 #align probability_theory.kernel.set_lintegral_deterministic ProbabilityTheory.kernel.set_lintegral_deterministic
 
 theorem integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : StronglyMeasurable f) : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    (hf : StronglyMeasurable f) : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac' _ _ hf]
 #align probability_theory.kernel.integral_deterministic' ProbabilityTheory.kernel.integral_deterministic'
 
 @[simp]
 theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    [MeasurableSingletonClass β] : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac _ (g a)]
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
 
 theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
-    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
 #align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
 
@@ -439,7 +439,7 @@ theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedS
 theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
-    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac f _ s]
 #align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
 
@@ -481,21 +481,21 @@ instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure
 #print ProbabilityTheory.kernel.lintegral_const /-
 @[simp]
 theorem lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} :
-    (∫⁻ x, f x ∂kernel.const α μ a) = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
+    ∫⁻ x, f x ∂kernel.const α μ a = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.lintegral_const ProbabilityTheory.kernel.lintegral_const
 -/
 
 #print ProbabilityTheory.kernel.set_lintegral_const /-
 @[simp]
 theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {s : Set β} :
-    (∫⁻ x in s, f x ∂kernel.const α μ a) = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
+    ∫⁻ x in s, f x ∂kernel.const α μ a = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
 -/
 
 #print ProbabilityTheory.kernel.integral_const /-
 @[simp]
 theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-    {f : β → E} {μ : Measure β} {a : α} : (∫ x, f x ∂kernel.const α μ a) = ∫ x, f x ∂μ := by
+    {f : β → E} {μ : Measure β} {a : α} : ∫ x, f x ∂kernel.const α μ a = ∫ x, f x ∂μ := by
   rw [kernel.const_apply]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 -/
@@ -504,7 +504,7 @@ theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [
 @[simp]
 theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
-    (∫ x in s, f x ∂kernel.const α μ a) = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
+    ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
 -/
 
@@ -554,19 +554,19 @@ theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ := by ext1 a;
 
 @[simp]
 theorem lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞) :
-    (∫⁻ b, f b ∂kernel.restrict κ hs a) = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
+    ∫⁻ b, f b ∂kernel.restrict κ hs a = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
 #align probability_theory.kernel.lintegral_restrict ProbabilityTheory.kernel.lintegral_restrict
 
 @[simp]
 theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞)
-    (t : Set β) : (∫⁻ b in t, f b ∂kernel.restrict κ hs a) = ∫⁻ b in t ∩ s, f b ∂κ a := by
+    (t : Set β) : ∫⁻ b in t, f b ∂kernel.restrict κ hs a = ∫⁻ b in t ∩ s, f b ∂κ a := by
   rw [restrict_apply, measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
 
 @[simp]
 theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
-    (∫ x in t, f x ∂kernel.restrict κ hs a) = ∫ x in t ∩ s, f x ∂κ a := by
+    ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
 
@@ -704,26 +704,25 @@ instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKernel η] :
 #align probability_theory.kernel.is_s_finite_kernel.piecewise ProbabilityTheory.kernel.IsSFiniteKernel.piecewise
 
 theorem lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
-    (∫⁻ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
+    ∫⁻ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.lintegral_piecewise ProbabilityTheory.kernel.lintegral_piecewise
 
 theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
-    (∫⁻ b in t, g b ∂piecewise hs κ η a) =
+    ∫⁻ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫⁻ b in t, g b ∂κ a else ∫⁻ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
 
 theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     (a : α) (g : β → E) :
-    (∫ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
+    ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
 theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
-    (∫ b in t, g b ∂piecewise hs κ η a) =
-      if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
+    ∫ b in t, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
 

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

@@ -56,6 +56,7 @@ open scoped MeasureTheory ENNReal NNReal BigOperators
 
 namespace ProbabilityTheory
 
+#print ProbabilityTheory.kernel /-
 /-- A kernel from a measurable space `α` to another measurable space `β` is a measurable function
 `κ : α → measure β`. The measurable space structure on `measure β` is given by
 `measure_theory.measure.measurable_space`. A map `κ : α → measure β` is measurable iff
@@ -66,6 +67,7 @@ def kernel (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] : AddSubmo
   zero_mem' := measurable_zero
   add_mem' f g hf hg := Measurable.add hf hg
 #align probability_theory.kernel ProbabilityTheory.kernel
+-/
 
 instance {α β : Type _} [MeasurableSpace α] [MeasurableSpace β] :
     CoeFun (kernel α β) fun _ => α → Measure β :=
@@ -97,10 +99,12 @@ def coeAddHom (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :
 
 include mα mβ
 
+#print ProbabilityTheory.kernel.zero_apply /-
 @[simp]
 theorem zero_apply (a : α) : (0 : kernel α β) a = 0 :=
   rfl
 #align probability_theory.kernel.zero_apply ProbabilityTheory.kernel.zero_apply
+-/
 
 @[simp]
 theorem coe_finset_sum (I : Finset ι) (κ : ι → kernel α β) : ⇑(∑ i in I, κ i) = ∑ i in I, κ i :=
@@ -148,11 +152,13 @@ theorem kernel.measure_le_bound (κ : kernel α β) [h : IsFiniteKernel κ] (a :
   (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
 #align probability_theory.kernel.measure_le_bound ProbabilityTheory.kernel.measure_le_bound
 
+#print ProbabilityTheory.isFiniteKernel_zero /-
 instance isFiniteKernel_zero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
     IsFiniteKernel (0 : kernel α β) :=
   ⟨⟨0, ENNReal.coe_lt_top, fun a => by
       simp only [kernel.zero_apply, measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
 #align probability_theory.is_finite_kernel_zero ProbabilityTheory.isFiniteKernel_zero
+-/
 
 instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :
     IsFiniteKernel (κ + η) :=
@@ -378,10 +384,12 @@ theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : S
   simp_rw [measure.dirac_apply' _ hs]
 #align probability_theory.kernel.deterministic_apply' ProbabilityTheory.kernel.deterministic_apply'
 
+#print ProbabilityTheory.kernel.isMarkovKernel_deterministic /-
 instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
     IsMarkovKernel (deterministic f hf) :=
   ⟨fun a => by rw [deterministic_apply hf]; infer_instance⟩
 #align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
+-/
 
 theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     (hf : Measurable f) : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
@@ -450,41 +458,55 @@ def const (α : Type _) {β : Type _} [MeasurableSpace α] {mβ : MeasurableSpac
 
 include mα mβ
 
+#print ProbabilityTheory.kernel.const_apply /-
 theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
   rfl
 #align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
+-/
 
+#print ProbabilityTheory.kernel.isFiniteKernel_const /-
 instance isFiniteKernel_const {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun a => le_rfl⟩⟩
 #align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernel_const
+-/
 
+#print ProbabilityTheory.kernel.isMarkovKernel_const /-
 instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
     IsMarkovKernel (const α μβ) :=
   ⟨fun a => hμβ⟩
 #align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernel_const
+-/
 
+#print ProbabilityTheory.kernel.lintegral_const /-
 @[simp]
 theorem lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} :
     (∫⁻ x, f x ∂kernel.const α μ a) = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.lintegral_const ProbabilityTheory.kernel.lintegral_const
+-/
 
+#print ProbabilityTheory.kernel.set_lintegral_const /-
 @[simp]
 theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {s : Set β} :
     (∫⁻ x in s, f x ∂kernel.const α μ a) = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
+-/
 
+#print ProbabilityTheory.kernel.integral_const /-
 @[simp]
 theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} {μ : Measure β} {a : α} : (∫ x, f x ∂kernel.const α μ a) = ∫ x, f x ∂μ := by
   rw [kernel.const_apply]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
+-/
 
+#print ProbabilityTheory.kernel.set_integral_const /-
 @[simp]
 theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
     (∫ x in s, f x ∂kernel.const α μ a) = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
+-/
 
 end Const
 

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

@@ -119,7 +119,7 @@ end Kernel
 
 /-- A kernel is a Markov kernel if every measure in its image is a probability measure. -/
 class IsMarkovKernel (κ : kernel α β) : Prop where
-  ProbabilityMeasure : ∀ a, ProbabilityMeasure (κ a)
+  IsProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
 #align probability_theory.is_markov_kernel ProbabilityTheory.IsMarkovKernel
 
 /-- A kernel is finite if every measure in its image is finite, with a uniform bound. -/
@@ -168,13 +168,13 @@ instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFinite
 variable {κ : kernel α β}
 
 instance IsMarkovKernel.is_probability_measure' [h : IsMarkovKernel κ] (a : α) :
-    ProbabilityMeasure (κ a) :=
-  IsMarkovKernel.probabilityMeasure a
+    IsProbabilityMeasure (κ a) :=
+  IsMarkovKernel.isProbabilityMeasure a
 #align probability_theory.is_markov_kernel.is_probability_measure' ProbabilityTheory.IsMarkovKernel.is_probability_measure'
 
-instance IsFiniteKernel.finiteMeasure [h : IsFiniteKernel κ] (a : α) : FiniteMeasure (κ a) :=
+instance IsFiniteKernel.isFiniteMeasure [h : IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
   ⟨(kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
-#align probability_theory.is_finite_kernel.is_finite_measure ProbabilityTheory.IsFiniteKernel.finiteMeasure
+#align probability_theory.is_finite_kernel.is_finite_measure ProbabilityTheory.IsFiniteKernel.isFiniteMeasure
 
 instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ] :
     IsFiniteKernel κ :=
@@ -314,13 +314,13 @@ instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
 theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
     (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
   classical
-    induction' I using Finset.induction with i I hi_nmem_I h_ind h
-    · rw [Finset.sum_empty]; infer_instance
-    · rw [Finset.sum_insert hi_nmem_I]
-      haveI : is_s_finite_kernel (κs i) := h i (Finset.mem_insert_self _ _)
-      have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
-        h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
-      exact is_s_finite_kernel.add _ _
+  induction' I using Finset.induction with i I hi_nmem_I h_ind h
+  · rw [Finset.sum_empty]; infer_instance
+  · rw [Finset.sum_insert hi_nmem_I]
+    haveI : is_s_finite_kernel (κs i) := h i (Finset.mem_insert_self _ _)
+    have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
+      h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
+    exact is_s_finite_kernel.add _ _
 #align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i m) -/
@@ -454,12 +454,12 @@ theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
   rfl
 #align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
 
-instance isFiniteKernel_const {μβ : Measure β} [hμβ : FiniteMeasure μβ] :
+instance isFiniteKernel_const {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun a => le_rfl⟩⟩
 #align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernel_const
 
-instance isMarkovKernel_const {μβ : Measure β} [hμβ : ProbabilityMeasure μβ] :
+instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
     IsMarkovKernel (const α μβ) :=
   ⟨fun a => hμβ⟩
 #align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernel_const

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

@@ -200,7 +200,7 @@ theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f
     κ = η := by
   ext (a s hs)
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
-  simp_rw [lintegral_indicator_const hs, one_mul] at h
+  simp_rw [lintegral_indicator_const hs, one_mul] at h 
   rw [h]
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 

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

@@ -52,7 +52,7 @@ Particular kernels:
 
 open MeasureTheory
 
-open MeasureTheory ENNReal NNReal BigOperators
+open scoped MeasureTheory ENNReal NNReal BigOperators
 
 namespace ProbabilityTheory
 

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

@@ -184,11 +184,7 @@ instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ]
 namespace Kernel
 
 @[ext]
-theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η :=
-  by
-  ext1
-  ext1 a
-  exact h a
+theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η := by ext1; ext1 a; exact h a
 #align probability_theory.kernel.ext ProbabilityTheory.kernel.ext
 
 theorem ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a :=
@@ -252,17 +248,12 @@ theorem sum_zero [Countable ι] : (kernel.sum fun i : ι => (0 : kernel α β))
 #align probability_theory.kernel.sum_zero ProbabilityTheory.kernel.sum_zero
 
 theorem sum_comm [Countable ι] (κ : ι → ι → kernel α β) :
-    (kernel.sum fun n => kernel.sum (κ n)) = kernel.sum fun m => kernel.sum fun n => κ n m :=
-  by
-  ext (a s hs)
-  simp_rw [sum_apply]
-  rw [measure.sum_comm]
+    (kernel.sum fun n => kernel.sum (κ n)) = kernel.sum fun m => kernel.sum fun n => κ n m := by
+  ext (a s hs); simp_rw [sum_apply]; rw [measure.sum_comm]
 #align probability_theory.kernel.sum_comm ProbabilityTheory.kernel.sum_comm
 
 @[simp]
-theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i :=
-  by
-  ext (a s hs)
+theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i := by ext (a s hs);
   simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
 #align probability_theory.kernel.sum_fintype ProbabilityTheory.kernel.sum_fintype
 
@@ -285,15 +276,11 @@ class ProbabilityTheory.IsSFiniteKernel (κ : kernel α β) : Prop where
 
 instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
     IsSFiniteKernel κ :=
-  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by
-      split_ifs
-      exact h
-      infer_instance, by
+  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by split_ifs; exact h; infer_instance,
+      by
       ext (a s hs)
       rw [kernel.sum_apply' _ _ hs]
-      have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 :=
-        by
-        ext1 i
+      have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by ext1 i;
         split_ifs <;> rfl
       rw [this, tsum_ite_eq]⟩⟩
 #align probability_theory.kernel.is_finite_kernel.is_s_finite_kernel ProbabilityTheory.kernel.IsFiniteKernel.isSFiniteKernel
@@ -328,8 +315,7 @@ theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
     (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
   classical
     induction' I using Finset.induction with i I hi_nmem_I h_ind h
-    · rw [Finset.sum_empty]
-      infer_instance
+    · rw [Finset.sum_empty]; infer_instance
     · rw [Finset.sum_insert hi_nmem_I]
       haveI : is_s_finite_kernel (κs i) := h i (Finset.mem_insert_self _ _)
       have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
@@ -394,9 +380,7 @@ theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : S
 
 instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
     IsMarkovKernel (deterministic f hf) :=
-  ⟨fun a => by
-    rw [deterministic_apply hf]
-    infer_instance⟩
+  ⟨fun a => by rw [deterministic_apply hf]; infer_instance⟩
 #align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
 
 theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
@@ -542,9 +526,7 @@ theorem restrict_apply' (κ : kernel α β) (hs : MeasurableSet s) (a : α) (ht
 #align probability_theory.kernel.restrict_apply' ProbabilityTheory.kernel.restrict_apply'
 
 @[simp]
-theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ :=
-  by
-  ext1 a
+theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ := by ext1 a;
   rw [kernel.restrict_apply, measure.restrict_univ]
 #align probability_theory.kernel.restrict_univ ProbabilityTheory.kernel.restrict_univ
 
@@ -672,16 +654,12 @@ theorem piecewise_apply (a : α) : piecewise hs κ η a = if a ∈ s then κ a e
 #align probability_theory.kernel.piecewise_apply ProbabilityTheory.kernel.piecewise_apply
 
 theorem piecewise_apply' (a : α) (t : Set β) :
-    piecewise hs κ η a t = if a ∈ s then κ a t else η a t :=
-  by
-  rw [piecewise_apply]
+    piecewise hs κ η a t = if a ∈ s then κ a t else η a t := by rw [piecewise_apply];
   split_ifs <;> rfl
 #align probability_theory.kernel.piecewise_apply' ProbabilityTheory.kernel.piecewise_apply'
 
 instance IsMarkovKernel.piecewise [IsMarkovKernel κ] [IsMarkovKernel η] :
-    IsMarkovKernel (piecewise hs κ η) :=
-  by
-  refine' ⟨fun a => ⟨_⟩⟩
+    IsMarkovKernel (piecewise hs κ η) := by refine' ⟨fun a => ⟨_⟩⟩;
   rw [piecewise_apply', measure_univ, measure_univ, if_t_t]
 #align probability_theory.kernel.is_markov_kernel.piecewise ProbabilityTheory.kernel.IsMarkovKernel.piecewise
 
@@ -704,35 +682,27 @@ instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKernel η] :
 #align probability_theory.kernel.is_s_finite_kernel.piecewise ProbabilityTheory.kernel.IsSFiniteKernel.piecewise
 
 theorem lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
-    (∫⁻ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a :=
-  by
-  simp_rw [piecewise_apply]
-  split_ifs <;> rfl
+    (∫⁻ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
+  simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.lintegral_piecewise ProbabilityTheory.kernel.lintegral_piecewise
 
 theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
     (∫⁻ b in t, g b ∂piecewise hs κ η a) =
       if a ∈ s then ∫⁻ b in t, g b ∂κ a else ∫⁻ b in t, g b ∂η a :=
-  by
-  simp_rw [piecewise_apply]
-  split_ifs <;> rfl
+  by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
 
 theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     (a : α) (g : β → E) :
-    (∫ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a :=
-  by
-  simp_rw [piecewise_apply]
-  split_ifs <;> rfl
+    (∫ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
+  simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
 theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
     (∫ b in t, g b ∂piecewise hs κ η a) =
       if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
-  by
-  simp_rw [piecewise_apply]
-  split_ifs <;> rfl
+  by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
 
 end Piecewise

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

@@ -231,7 +231,7 @@ protected noncomputable def sum [Countable ι] (κ : ι → kernel α β) : kern
   property := by
     refine' measure.measurable_of_measurable_coe _ fun s hs => _
     simp_rw [measure.sum_apply _ hs]
-    exact Measurable.eNNReal_tsum fun n => kernel.measurable_coe (κ n) hs
+    exact Measurable.ennreal_tsum fun n => kernel.measurable_coe (κ n) hs
 #align probability_theory.kernel.sum ProbabilityTheory.kernel.sum
 
 theorem sum_apply [Countable ι] (κ : ι → kernel α β) (a : α) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,11 +4,12 @@ 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.basic
-! leanprover-community/mathlib commit a9545e8a564bac7f24637443f52ae955474e4991
+! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.MeasureTheory.Constructions.Prod
+import Mathbin.MeasureTheory.Integral.Bochner
+import Mathbin.MeasureTheory.Constructions.Prod.Basic
 
 /-!
 # Markov Kernels

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

@@ -118,7 +118,7 @@ end Kernel
 
 /-- A kernel is a Markov kernel if every measure in its image is a probability measure. -/
 class IsMarkovKernel (κ : kernel α β) : Prop where
-  IsProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
+  ProbabilityMeasure : ∀ a, ProbabilityMeasure (κ a)
 #align probability_theory.is_markov_kernel ProbabilityTheory.IsMarkovKernel
 
 /-- A kernel is finite if every measure in its image is finite, with a uniform bound. -/
@@ -167,13 +167,13 @@ instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFinite
 variable {κ : kernel α β}
 
 instance IsMarkovKernel.is_probability_measure' [h : IsMarkovKernel κ] (a : α) :
-    IsProbabilityMeasure (κ a) :=
-  IsMarkovKernel.isProbabilityMeasure a
+    ProbabilityMeasure (κ a) :=
+  IsMarkovKernel.probabilityMeasure a
 #align probability_theory.is_markov_kernel.is_probability_measure' ProbabilityTheory.IsMarkovKernel.is_probability_measure'
 
-instance IsFiniteKernel.isFiniteMeasure [h : IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
+instance IsFiniteKernel.finiteMeasure [h : IsFiniteKernel κ] (a : α) : FiniteMeasure (κ a) :=
   ⟨(kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
-#align probability_theory.is_finite_kernel.is_finite_measure ProbabilityTheory.IsFiniteKernel.isFiniteMeasure
+#align probability_theory.is_finite_kernel.is_finite_measure ProbabilityTheory.IsFiniteKernel.finiteMeasure
 
 instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ] :
     IsFiniteKernel κ :=
@@ -469,12 +469,12 @@ theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
   rfl
 #align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
 
-instance isFiniteKernel_const {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
+instance isFiniteKernel_const {μβ : Measure β} [hμβ : FiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun a => le_rfl⟩⟩
 #align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernel_const
 
-instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
+instance isMarkovKernel_const {μβ : Measure β} [hμβ : ProbabilityMeasure μβ] :
     IsMarkovKernel (const α μβ) :=
   ⟨fun a => hμβ⟩
 #align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernel_const

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

@@ -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.basic
-! leanprover-community/mathlib commit 483dd86cfec4a1380d22b1f6acd4c3dc53f501ff
+! leanprover-community/mathlib commit a9545e8a564bac7f24637443f52ae955474e4991
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -30,7 +30,7 @@ Classes of kernels:
   particular that all measures in the image of `κ` are finite, but is stronger since it requires an
   uniform bound. This stronger condition is necessary to ensure that the composition of two finite
   kernels is finite.
-* `probability_theory.kernel.is_s_finite_kernel κ`: a kernel is called s-finite if it is a countable
+* `probability_theory.is_s_finite_kernel κ`: a kernel is called s-finite if it is a countable
   sum of finite kernels.
 
 Particular kernels:
@@ -183,14 +183,23 @@ instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ]
 namespace Kernel
 
 @[ext]
-theorem ext {κ : kernel α β} {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η :=
+theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η :=
   by
   ext1
   ext1 a
   exact h a
 #align probability_theory.kernel.ext ProbabilityTheory.kernel.ext
 
-theorem ext_fun {κ η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a) :
+theorem ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a :=
+  ⟨fun h a => by rw [h], ext⟩
+#align probability_theory.kernel.ext_iff ProbabilityTheory.kernel.ext_iff
+
+theorem ext_iff' {η : kernel α β} :
+    κ = η ↔ ∀ (a) (s : Set β) (hs : MeasurableSet s), κ a s = η a s := by
+  simp_rw [ext_iff, measure.ext_iff]
+#align probability_theory.kernel.ext_iff' ProbabilityTheory.kernel.ext_iff'
+
+theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a) :
     κ = η := by
   ext (a s hs)
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
@@ -198,6 +207,11 @@ theorem ext_fun {κ η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b,
   rw [h]
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 
+theorem ext_fun_iff {η : kernel α β} :
+    κ = η ↔ ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a :=
+  ⟨fun h a f hf => by rw [h], ext_fun⟩
+#align probability_theory.kernel.ext_fun_iff ProbabilityTheory.kernel.ext_fun_iff
+
 protected theorem measurable (κ : kernel α β) : Measurable κ :=
   κ.Prop
 #align probability_theory.kernel.measurable ProbabilityTheory.kernel.measurable
@@ -264,9 +278,9 @@ end Sum
 section SFinite
 
 /-- A kernel is s-finite if it can be written as the sum of countably many finite kernels. -/
-class IsSFiniteKernel (κ : kernel α β) : Prop where
+class ProbabilityTheory.IsSFiniteKernel (κ : kernel α β) : Prop where
   tsum_finite : ∃ κs : ℕ → kernel α β, (∀ n, IsFiniteKernel (κs n)) ∧ κ = kernel.sum κs
-#align probability_theory.kernel.is_s_finite_kernel ProbabilityTheory.kernel.IsSFiniteKernel
+#align probability_theory.is_s_finite_kernel ProbabilityTheory.IsSFiniteKernel
 
 instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
     IsSFiniteKernel κ :=
@@ -384,6 +398,58 @@ instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
     infer_instance⟩
 #align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
 
+theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
+    (hf : Measurable f) : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+  rw [kernel.deterministic_apply, lintegral_dirac' _ hf]
+#align probability_theory.kernel.lintegral_deterministic' ProbabilityTheory.kernel.lintegral_deterministic'
+
+@[simp]
+theorem lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
+    [MeasurableSingletonClass β] : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+  rw [kernel.deterministic_apply, lintegral_dirac (g a) f]
+#align probability_theory.kernel.lintegral_deterministic ProbabilityTheory.kernel.lintegral_deterministic
+
+theorem set_lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
+    (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
+    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+  rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]
+#align probability_theory.kernel.set_lintegral_deterministic' ProbabilityTheory.kernel.set_lintegral_deterministic'
+
+@[simp]
+theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
+    [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
+    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+  rw [kernel.deterministic_apply, set_lintegral_dirac f s]
+#align probability_theory.kernel.set_lintegral_deterministic ProbabilityTheory.kernel.set_lintegral_deterministic
+
+theorem integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
+    (hf : StronglyMeasurable f) : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+  rw [kernel.deterministic_apply, integral_dirac' _ _ hf]
+#align probability_theory.kernel.integral_deterministic' ProbabilityTheory.kernel.integral_deterministic'
+
+@[simp]
+theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
+    [MeasurableSingletonClass β] : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+  rw [kernel.deterministic_apply, integral_dirac _ (g a)]
+#align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
+
+theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
+    (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
+    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+  rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
+#align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
+
+@[simp]
+theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
+    [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
+    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+  rw [kernel.deterministic_apply, set_integral_dirac f _ s]
+#align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
+
 end Deterministic
 
 section Const
@@ -413,6 +479,28 @@ instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure
   ⟨fun a => hμβ⟩
 #align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernel_const
 
+@[simp]
+theorem lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} :
+    (∫⁻ x, f x ∂kernel.const α μ a) = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
+#align probability_theory.kernel.lintegral_const ProbabilityTheory.kernel.lintegral_const
+
+@[simp]
+theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {s : Set β} :
+    (∫⁻ x in s, f x ∂kernel.const α μ a) = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
+#align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
+
+@[simp]
+theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    {f : β → E} {μ : Measure β} {a : α} : (∫ x, f x ∂kernel.const α μ a) = ∫ x, f x ∂μ := by
+  rw [kernel.const_apply]
+#align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
+
+@[simp]
+theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
+    (∫ x in s, f x ∂kernel.const α μ a) = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
+#align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
+
 end Const
 
 omit mα
@@ -452,10 +540,31 @@ theorem restrict_apply' (κ : kernel α β) (hs : MeasurableSet s) (a : α) (ht
   rw [restrict_apply κ hs a, measure.restrict_apply ht]
 #align probability_theory.kernel.restrict_apply' ProbabilityTheory.kernel.restrict_apply'
 
+@[simp]
+theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ :=
+  by
+  ext1 a
+  rw [kernel.restrict_apply, measure.restrict_univ]
+#align probability_theory.kernel.restrict_univ ProbabilityTheory.kernel.restrict_univ
+
+@[simp]
 theorem lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞) :
     (∫⁻ b, f b ∂kernel.restrict κ hs a) = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
 #align probability_theory.kernel.lintegral_restrict ProbabilityTheory.kernel.lintegral_restrict
 
+@[simp]
+theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞)
+    (t : Set β) : (∫⁻ b in t, f b ∂kernel.restrict κ hs a) = ∫⁻ b in t ∩ s, f b ∂κ a := by
+  rw [restrict_apply, measure.restrict_restrict' hs]
+#align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
+
+@[simp]
+theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
+    (∫ x in t, f x ∂kernel.restrict κ hs a) = ∫ x in t ∩ s, f x ∂κ a := by
+  rw [restrict_apply, measure.restrict_restrict' hs]
+#align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
+
 instance IsFiniteKernel.restrict (κ : kernel α β) [IsFiniteKernel κ] (hs : MeasurableSet s) :
     IsFiniteKernel (kernel.restrict κ hs) :=
   by
@@ -474,6 +583,159 @@ instance IsSFiniteKernel.restrict (κ : kernel α β) [IsSFiniteKernel κ] (hs :
 
 end Restrict
 
+section ComapRight
+
+variable {γ : Type _} {mγ : MeasurableSpace γ} {f : γ → β}
+
+include mγ
+
+/-- Kernel with value `(κ a).comap f`, for a measurable embedding `f`. That is, for a measurable set
+`t : set β`, `comap_right κ hf a t = κ a (f '' t)`. -/
+noncomputable def comapRight (κ : kernel α β) (hf : MeasurableEmbedding f) : kernel α γ
+    where
+  val a := (κ a).comap f
+  property := by
+    refine' measure.measurable_measure.mpr fun t ht => _
+    have : (fun a => measure.comap f (κ a) t) = fun a => κ a (f '' t) :=
+      by
+      ext1 a
+      rw [measure.comap_apply _ hf.injective (fun s' hs' => _) _ ht]
+      exact hf.measurable_set_image.mpr hs'
+    rw [this]
+    exact kernel.measurable_coe _ (hf.measurable_set_image.mpr ht)
+#align probability_theory.kernel.comap_right ProbabilityTheory.kernel.comapRight
+
+theorem comapRight_apply (κ : kernel α β) (hf : MeasurableEmbedding f) (a : α) :
+    comapRight κ hf a = Measure.comap f (κ a) :=
+  rfl
+#align probability_theory.kernel.comap_right_apply ProbabilityTheory.kernel.comapRight_apply
+
+theorem comapRight_apply' (κ : kernel α β) (hf : MeasurableEmbedding f) (a : α) {t : Set γ}
+    (ht : MeasurableSet t) : comapRight κ hf a t = κ a (f '' t) := by
+  rw [comap_right_apply,
+    measure.comap_apply _ hf.injective (fun s => hf.measurable_set_image.mpr) _ ht]
+#align probability_theory.kernel.comap_right_apply' ProbabilityTheory.kernel.comapRight_apply'
+
+theorem IsMarkovKernel.comapRight (κ : kernel α β) (hf : MeasurableEmbedding f)
+    (hκ : ∀ a, κ a (Set.range f) = 1) : IsMarkovKernel (comapRight κ hf) :=
+  by
+  refine' ⟨fun a => ⟨_⟩⟩
+  rw [comap_right_apply' κ hf a MeasurableSet.univ]
+  simp only [Set.image_univ, Subtype.range_coe_subtype, Set.setOf_mem_eq]
+  exact hκ a
+#align probability_theory.kernel.is_markov_kernel.comap_right ProbabilityTheory.kernel.IsMarkovKernel.comapRight
+
+instance IsFiniteKernel.comapRight (κ : kernel α β) [IsFiniteKernel κ]
+    (hf : MeasurableEmbedding f) : IsFiniteKernel (comapRight κ hf) :=
+  by
+  refine' ⟨⟨is_finite_kernel.bound κ, is_finite_kernel.bound_lt_top κ, fun a => _⟩⟩
+  rw [comap_right_apply' κ hf a MeasurableSet.univ]
+  exact measure_le_bound κ a _
+#align probability_theory.kernel.is_finite_kernel.comap_right ProbabilityTheory.kernel.IsFiniteKernel.comapRight
+
+instance IsSFiniteKernel.comapRight (κ : kernel α β) [IsSFiniteKernel κ]
+    (hf : MeasurableEmbedding f) : IsSFiniteKernel (comapRight κ hf) :=
+  by
+  refine' ⟨⟨fun n => comap_right (seq κ n) hf, inferInstance, _⟩⟩
+  ext1 a
+  rw [sum_apply]
+  simp_rw [comap_right_apply _ hf]
+  have :
+    (measure.sum fun n => measure.comap f (seq κ n a)) =
+      measure.comap f (measure.sum fun n => seq κ n a) :=
+    by
+    ext1 t ht
+    rw [measure.comap_apply _ hf.injective (fun s' => hf.measurable_set_image.mpr) _ ht,
+      measure.sum_apply _ ht, measure.sum_apply _ (hf.measurable_set_image.mpr ht)]
+    congr with n : 1
+    rw [measure.comap_apply _ hf.injective (fun s' => hf.measurable_set_image.mpr) _ ht]
+  rw [this, measure_sum_seq]
+#align probability_theory.kernel.is_s_finite_kernel.comap_right ProbabilityTheory.kernel.IsSFiniteKernel.comapRight
+
+end ComapRight
+
+section Piecewise
+
+variable {η : kernel α β} {s : Set α} {hs : MeasurableSet s} [DecidablePred (· ∈ s)]
+
+/-- `piecewise hs κ η` is the kernel equal to `κ` on the measurable set `s` and to `η` on its
+complement. -/
+def piecewise (hs : MeasurableSet s) (κ η : kernel α β) : kernel α β
+    where
+  val a := if a ∈ s then κ a else η a
+  property := Measurable.piecewise hs (kernel.measurable _) (kernel.measurable _)
+#align probability_theory.kernel.piecewise ProbabilityTheory.kernel.piecewise
+
+theorem piecewise_apply (a : α) : piecewise hs κ η a = if a ∈ s then κ a else η a :=
+  rfl
+#align probability_theory.kernel.piecewise_apply ProbabilityTheory.kernel.piecewise_apply
+
+theorem piecewise_apply' (a : α) (t : Set β) :
+    piecewise hs κ η a t = if a ∈ s then κ a t else η a t :=
+  by
+  rw [piecewise_apply]
+  split_ifs <;> rfl
+#align probability_theory.kernel.piecewise_apply' ProbabilityTheory.kernel.piecewise_apply'
+
+instance IsMarkovKernel.piecewise [IsMarkovKernel κ] [IsMarkovKernel η] :
+    IsMarkovKernel (piecewise hs κ η) :=
+  by
+  refine' ⟨fun a => ⟨_⟩⟩
+  rw [piecewise_apply', measure_univ, measure_univ, if_t_t]
+#align probability_theory.kernel.is_markov_kernel.piecewise ProbabilityTheory.kernel.IsMarkovKernel.piecewise
+
+instance IsFiniteKernel.piecewise [IsFiniteKernel κ] [IsFiniteKernel η] :
+    IsFiniteKernel (piecewise hs κ η) :=
+  by
+  refine' ⟨⟨max (is_finite_kernel.bound κ) (is_finite_kernel.bound η), _, fun a => _⟩⟩
+  · exact max_lt (is_finite_kernel.bound_lt_top κ) (is_finite_kernel.bound_lt_top η)
+  rw [piecewise_apply']
+  exact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _))
+#align probability_theory.kernel.is_finite_kernel.piecewise ProbabilityTheory.kernel.IsFiniteKernel.piecewise
+
+instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    IsSFiniteKernel (piecewise hs κ η) :=
+  by
+  refine' ⟨⟨fun n => piecewise hs (seq κ n) (seq η n), inferInstance, _⟩⟩
+  ext1 a
+  simp_rw [sum_apply, kernel.piecewise_apply]
+  split_ifs <;> exact (measure_sum_seq _ a).symm
+#align probability_theory.kernel.is_s_finite_kernel.piecewise ProbabilityTheory.kernel.IsSFiniteKernel.piecewise
+
+theorem lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
+    (∫⁻ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a :=
+  by
+  simp_rw [piecewise_apply]
+  split_ifs <;> rfl
+#align probability_theory.kernel.lintegral_piecewise ProbabilityTheory.kernel.lintegral_piecewise
+
+theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
+    (∫⁻ b in t, g b ∂piecewise hs κ η a) =
+      if a ∈ s then ∫⁻ b in t, g b ∂κ a else ∫⁻ b in t, g b ∂η a :=
+  by
+  simp_rw [piecewise_apply]
+  split_ifs <;> rfl
+#align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
+
+theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    (a : α) (g : β → E) :
+    (∫ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a :=
+  by
+  simp_rw [piecewise_apply]
+  split_ifs <;> rfl
+#align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
+
+theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
+    (∫ b in t, g b ∂piecewise hs κ η a) =
+      if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
+  by
+  simp_rw [piecewise_apply]
+  split_ifs <;> rfl
+#align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
+
+end Piecewise
+
 end Kernel
 
 end ProbabilityTheory

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

@@ -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.basic
-! leanprover-community/mathlib commit caf83ba4dfbf4e2f28e1ae6a0780cbafc3d19d6f
+! leanprover-community/mathlib commit 483dd86cfec4a1380d22b1f6acd4c3dc53f501ff
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -21,36 +21,30 @@ sets `s` of `β`, `a ↦ κ a s` is measurable.
 ## Main definitions
 
 Classes of kernels:
-* `kernel α β`: kernels from `α` to `β`, defined as the `add_submonoid` of the measurable
-  functions in `α → measure β`.
-* `is_markov_kernel κ`: a kernel from `α` to `β` is said to be a Markov kernel if for all `a : α`,
-  `k a` is a probability measure.
-* `is_finite_kernel κ`: a kernel from `α` to `β` is said to be finite if there exists `C : ℝ≥0∞`
-  such that `C < ∞` and for all `a : α`, `κ a univ ≤ C`. This implies in particular that all
-  measures in the image of `κ` are finite, but is stronger since it requires an uniform bound. This
-  stronger condition is necessary to ensure that the composition of two finite kernels is finite.
-* `is_s_finite_kernel κ`: a kernel is called s-finite if it is a countable sum of finite kernels.
+* `probability_theory.kernel α β`: kernels from `α` to `β`, defined as the `add_submonoid` of the
+  measurable functions in `α → measure β`.
+* `probability_theory.is_markov_kernel κ`: a kernel from `α` to `β` is said to be a Markov kernel
+  if for all `a : α`, `k a` is a probability measure.
+* `probability_theory.is_finite_kernel κ`: a kernel from `α` to `β` is said to be finite if there
+  exists `C : ℝ≥0∞` such that `C < ∞` and for all `a : α`, `κ a univ ≤ C`. This implies in
+  particular that all measures in the image of `κ` are finite, but is stronger since it requires an
+  uniform bound. This stronger condition is necessary to ensure that the composition of two finite
+  kernels is finite.
+* `probability_theory.kernel.is_s_finite_kernel κ`: a kernel is called s-finite if it is a countable
+  sum of finite kernels.
 
 Particular kernels:
-* `deterministic (f : α → β) (hf : measurable f)`: kernel `a ↦ measure.dirac (f a)`.
-* `const α (μβ : measure β)`: constant kernel `a ↦ μβ`.
-* `kernel.restrict κ (hs : measurable_set s)`: kernel for which the image of `a : α` is
-  `(κ a).restrict s`.
+* `probability_theory.kernel.deterministic (f : α → β) (hf : measurable f)`:
+  kernel `a ↦ measure.dirac (f a)`.
+* `probability_theory.kernel.const α (μβ : measure β)`: constant kernel `a ↦ μβ`.
+* `probability_theory.kernel.restrict κ (hs : measurable_set s)`: kernel for which the image of
+  `a : α` is `(κ a).restrict s`.
   Integral: `∫⁻ b, f b ∂(kernel.restrict κ hs a) = ∫⁻ b in s, f b ∂(κ a)`
-* `kernel.with_density κ (f : α → β → ℝ≥0∞)`: kernel `a ↦ (κ a).with_density (f a)`.
-  It is defined if `κ` is s-finite. If `f` is finite everywhere, then this is also an s-finite
-  kernel. The class of s-finite kernels is the smallest class of kernels that contains finite
-  kernels and which is stable by `with_density`.
-  Integral: `∫⁻ b, g b ∂(with_density κ f a) = ∫⁻ b, f a b * g b ∂(κ a)`
 
 ## Main statements
 
-* `ext_fun`: if `∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a)` for all measurable functions `f` and all `a`,
-  then the two kernels `κ` and `η` are equal.
-
-* `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.
+* `probability_theory.kernel.ext_fun`: if `∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a)` for all measurable
+  functions `f` and all `a`, then the two kernels `κ` and `η` are equal.
 
 -/
 
@@ -480,376 +474,6 @@ instance IsSFiniteKernel.restrict (κ : kernel α β) [IsSFiniteKernel κ] (hs :
 
 end Restrict
 
-section MeasurableLintegral
-
-/-- 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 } :=
-  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]
-  · -- 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 :=
-        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]
-      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' } :=
-      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_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' } :=
-      by
-      ext1 a
-      rw [← Set.diff_inter_self_eq_diff, Set.inter_univ, measure_diff]
-      · exact Set.subset_univ _
-      · exact (@measurable_prod_mk_left α β _ _ a) t' ht'
-      · exact measure_ne_top _ _
-    rw [this]
-    exact Measurable.sub (kernel.measurable_coe κ MeasurableSet.univ) h_meas
-  · -- 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 }) :=
-      by
-      ext1 a
-      congr with b
-      simp only [Set.mem_unionᵢ, Set.supᵢ_eq_unionᵢ, Set.mem_setOf_eq]
-      rfl
-    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 } :=
-      by
-      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
-        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)
-    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
-
-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 } :=
-  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)
-  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
-
-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 :=
-  by
-  simp_rw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]
-  exact Measurable.const_mul (measurable_prod_mk_mem _ 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 :=
-  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
-  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 _
-  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
-  · intro g₁ g₂ h_disj hm₁ hm₂
-    simp only [simple_func.coe_add, Pi.add_apply]
-    have h_add :
-      (fun a => ∫⁻ b, g₁ (a, b) + g₂ (a, b) ∂κ a) =
-        (fun a => ∫⁻ b, g₁ (a, b) ∂κ a) + fun a => ∫⁻ b, g₂ (a, b) ∂κ a :=
-      by
-      ext1 a
-      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
-
-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 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 :=
-  by
-  simp_rw [← lintegral_restrict κ hs]
-  exact measurable_lintegral _ hf
-#align probability_theory.kernel.measurable_set_lintegral ProbabilityTheory.kernel.measurableSet_lintegral
-
-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'
-
-end MeasurableLintegral
-
-section WithDensity
-
-variable {f : α → β → ℝ≥0∞}
-
-/-- Kernel with image `(κ a).with_density (f a)` if `function.uncurry f` is measurable, and
-with image 0 otherwise. If `function.uncurry f` is measurable, it satisfies
-`∫⁻ b, g b ∂(with_density κ f hf a) = ∫⁻ b, f a b * g b ∂(κ a)`. -/
-noncomputable def withDensity (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0∞) :
-    kernel α β :=
-  @dite _ (Measurable (Function.uncurry f)) (Classical.dec _)
-    (fun hf =>
-      ({  val := fun a => (κ a).withDensity (f a)
-          property := by
-            refine' measure.measurable_of_measurable_coe _ fun s hs => _
-            simp_rw [with_density_apply _ hs]
-            exact measurable_set_lintegral κ hf hs } :
-        kernel α β))
-    fun hf => 0
-#align probability_theory.kernel.with_density ProbabilityTheory.kernel.withDensity
-
-theorem withDensity_of_not_measurable (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical exact dif_neg hf
-#align probability_theory.kernel.with_density_of_not_measurable ProbabilityTheory.kernel.withDensity_of_not_measurable
-
-protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : Measurable (Function.uncurry f)) (a : α) : withDensity κ f a = (κ a).withDensity (f a) :=
-  by
-  classical
-    rw [with_density, dif_pos hf]
-    rfl
-#align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
-
-theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : Measurable (Function.uncurry f)) (a : α) {s : Set β} (hs : MeasurableSet s) :
-    withDensity κ f a s = ∫⁻ b in s, f a b ∂κ a := by
-  rw [kernel.with_density_apply κ hf, with_density_apply _ hs]
-#align probability_theory.kernel.with_density_apply' ProbabilityTheory.kernel.withDensity_apply'
-
-theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : Measurable (Function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
-    (∫⁻ b, g b ∂withDensity κ f a) = ∫⁻ b, f a b * g b ∂κ a :=
-  by
-  rw [kernel.with_density_apply _ hf,
-    lintegral_with_density_eq_lintegral_mul _ (Measurable.of_uncurry_left hf) hg]
-  simp_rw [Pi.mul_apply]
-#align probability_theory.kernel.lintegral_with_density ProbabilityTheory.kernel.lintegral_withDensity
-
-theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
-    (f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f :=
-  by
-  by_cases hf : Measurable (Function.uncurry f)
-  · ext (a s hs) : 2
-    simp only [kernel.with_density_apply _ hf, coe_fn_add, Pi.add_apply, with_density_add_measure,
-      measure.add_apply]
-  · simp_rw [with_density_of_not_measurable _ hf]
-    rw [zero_add]
-#align probability_theory.kernel.with_density_add_left ProbabilityTheory.kernel.withDensity_add_left
-
-theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
-    (f : α → β → ℝ≥0∞) :
-    @withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
-      kernel.sum fun i => withDensity (κ i) f :=
-  by
-  by_cases hf : Measurable (Function.uncurry f)
-  · ext1 a
-    simp_rw [sum_apply, kernel.with_density_apply _ hf, sum_apply,
-      with_density_sum (fun n => κ n a) (f a)]
-  · simp_rw [with_density_of_not_measurable _ hf]
-    exact sum_zero.symm
-#align probability_theory.kernel.with_density_kernel_sum ProbabilityTheory.kernel.withDensity_kernel_sum
-
-theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ] {f : ι → α → β → ℝ≥0∞}
-    (hf : ∀ i, Measurable (Function.uncurry (f i))) :
-    withDensity κ (∑' n, f n) = kernel.sum fun n => withDensity κ (f n) :=
-  by
-  have h_sum_a : ∀ a, Summable fun n => f n a := fun a => pi.summable.mpr fun b => ENNReal.summable
-  have h_sum : Summable fun n => f n := pi.summable.mpr h_sum_a
-  ext (a s hs) : 2
-  rw [sum_apply' _ a hs, with_density_apply' κ _ a hs]
-  swap
-  · have : Function.uncurry (∑' n, f n) = ∑' n, Function.uncurry (f n) :=
-      by
-      ext1 p
-      simp only [Function.uncurry_def]
-      rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]
-      exact pi.summable.mpr fun p => ENNReal.summable
-    rw [this]
-    exact Measurable.eNNReal_tsum' hf
-  have : (∫⁻ b in s, (∑' n, f n) a b ∂κ a) = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a :=
-    by
-    congr with b
-    rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
-  rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).AEMeasurable]
-  congr with n
-  rw [with_density_apply' _ (hf n) a hs]
-#align probability_theory.kernel.with_density_tsum ProbabilityTheory.kernel.withDensity_tsum
-
-/-- If a kernel `κ` is finite and a function `f : α → β → ℝ≥0∞` is bounded, then `with_density κ f`
-is finite. -/
-theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKernel κ] {B : ℝ≥0∞}
-    (hB_top : B ≠ ∞) (hf_B : ∀ a b, f a b ≤ B) : IsFiniteKernel (withDensity κ f) :=
-  by
-  by_cases hf : Measurable (Function.uncurry f)
-  ·
-    exact
-      ⟨⟨B * is_finite_kernel.bound κ, ENNReal.mul_lt_top hB_top (is_finite_kernel.bound_ne_top κ),
-          fun a => by
-          rw [with_density_apply' κ hf a MeasurableSet.univ]
-          calc
-            (∫⁻ b in Set.univ, f a b ∂κ a) ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
-            _ = B * κ a Set.univ := by simp only [measure.restrict_univ, lintegral_const]
-            _ ≤ B * is_finite_kernel.bound κ := mul_le_mul_left' (measure_le_bound κ a Set.univ) _
-            ⟩⟩
-  · rw [with_density_of_not_measurable _ hf]
-    infer_instance
-#align probability_theory.kernel.is_finite_kernel_with_density_of_bounded ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded
-
-/-- Auxiliary lemma for `is_s_finite_kernel_with_density`.
-If a kernel `κ` is finite, then `with_density κ f` is s-finite. -/
-theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFiniteKernel κ]
-    (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) :=
-  by
-  -- We already have that for `f` bounded from above and a `κ` a finite kernel,
-  -- `with_density κ f` is finite. We write any function as a countable sum of bounded
-  -- functions, and decompose an s-finite kernel as a sum of finite kernels. We then use that
-  -- `with_density` commutes with sums for both arguments and get a sum of finite kernels.
-  by_cases hf : Measurable (Function.uncurry f)
-  swap
-  · rw [with_density_of_not_measurable _ hf]
-    infer_instance
-  let fs : ℕ → α → β → ℝ≥0∞ := fun n a b => min (f a b) (n + 1) - min (f a b) n
-  have h_le : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ n :=
-    by
-    intro a b n hn
-    have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
-    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
-    · refine' this.trans (le_of_eq _)
-      rw [ENNReal.ofReal_coe_nat]
-    · norm_cast
-      exact zero_le _
-  have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 :=
-    by
-    intro a b n hn
-    suffices min (f a b) (n + 1) = f a b ∧ min (f a b) n = f a b by
-      simp_rw [fs, this.1, this.2, tsub_self (f a b)]
-    exact
-      ⟨min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),
-        min_eq_left (h_le a b n hn)⟩
-  have hf_eq_tsum : f = ∑' n, fs n :=
-    by
-    have h_sum_a : ∀ a, Summable fun n => fs n a :=
-      by
-      refine' fun a => pi.summable.mpr fun b => _
-      suffices : ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0
-      exact summable_of_ne_finset_zero this
-      intro n hn_not_mem
-      rw [Finset.mem_range, not_lt] at hn_not_mem
-      exact h_zero a b n hn_not_mem
-    ext (a b) : 2
-    rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
-      ENNReal.tsum_eq_liminf_sum_nat]
-    have h_finset_sum : ∀ n, (∑ i in Finset.range n, fs i a b) = min (f a b) n :=
-      by
-      intro n
-      induction' n with n hn
-      · simp only [Finset.range_zero, Finset.sum_empty, algebraMap.coe_zero, min_zero]
-      rw [Finset.sum_range_succ, hn]
-      simp_rw [fs]
-      norm_cast
-      rw [add_tsub_cancel_iff_le]
-      refine' min_le_min le_rfl _
-      norm_cast
-      exact Nat.le_succ n
-    simp_rw [h_finset_sum]
-    refine' (Filter.Tendsto.liminf_eq _).symm
-    refine' Filter.Tendsto.congr' _ tendsto_const_nhds
-    rw [Filter.EventuallyEq, Filter.eventually_atTop]
-    exact ⟨⌈(f a b).toReal⌉₊, fun n hn => (min_eq_left (h_le a b n hn)).symm⟩
-  rw [hf_eq_tsum, with_density_tsum _ fun n : ℕ => _]
-  swap
-  · exact (hf.min measurable_const).sub (hf.min measurable_const)
-  refine' is_s_finite_kernel_sum fun n => _
-  suffices is_finite_kernel (with_density κ (fs n))
-    by
-    haveI := this
-    infer_instance
-  refine' is_finite_kernel_with_density_of_bounded _ (ENNReal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => _
-  norm_cast
-  calc
-    fs n a b ≤ min (f a b) (n + 1) := tsub_le_self
-    _ ≤ n + 1 := (min_le_right _ _)
-    _ = ↑(n + 1) := by norm_cast
-    
-#align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
-
-/-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
-`with_density κ f` is s-finite. -/
-theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) :=
-  by
-  have h_eq_sum : with_density κ f = kernel.sum fun i => with_density (seq κ i) f :=
-    by
-    rw [← with_density_kernel_sum _ _]
-    congr
-    exact (kernel_sum_seq κ).symm
-  rw [h_eq_sum]
-  exact
-    is_s_finite_kernel_sum fun n =>
-      is_s_finite_kernel_with_density_of_is_finite_kernel (seq κ n) hf_ne_top
-#align probability_theory.kernel.is_s_finite_kernel.with_density ProbabilityTheory.kernel.IsSFiniteKernel.withDensity
-
-/-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0`, `with_density κ f` is s-finite. -/
-instance (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0) :
-    IsSFiniteKernel (withDensity κ fun a b => f a b) :=
-  IsSFiniteKernel.withDensity κ fun _ _ => ENNReal.coe_ne_top
-
-end WithDensity
-
 end Kernel
 
 end ProbabilityTheory

mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248

@@ -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.basic
-! leanprover-community/mathlib commit 97d1aa955750bd57a7eeef91de310e633881670b
+! leanprover-community/mathlib commit caf83ba4dfbf4e2f28e1ae6a0780cbafc3d19d6f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -32,7 +32,7 @@ Classes of kernels:
 * `is_s_finite_kernel κ`: a kernel is called s-finite if it is a countable sum of finite kernels.
 
 Particular kernels:
-* `deterministic {f : α → β} (hf : measurable f)`: kernel `a ↦ measure.dirac (f a)`.
+* `deterministic (f : α → β) (hf : measurable f)`: kernel `a ↦ measure.dirac (f a)`.
 * `const α (μβ : measure β)`: constant kernel `a ↦ μβ`.
 * `kernel.restrict κ (hs : measurable_set s)`: kernel for which the image of `a : α` is
   `(κ a).restrict s`.
@@ -361,7 +361,7 @@ end SFinite
 section Deterministic
 
 /-- Kernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel. -/
-noncomputable def deterministic {f : α → β} (hf : Measurable f) : kernel α β
+noncomputable def deterministic (f : α → β) (hf : Measurable f) : kernel α β
     where
   val a := Measure.dirac (f a)
   property := by
@@ -371,12 +371,12 @@ noncomputable def deterministic {f : α → β} (hf : Measurable f) : kernel α
 #align probability_theory.kernel.deterministic ProbabilityTheory.kernel.deterministic
 
 theorem deterministic_apply {f : α → β} (hf : Measurable f) (a : α) :
-    deterministic hf a = Measure.dirac (f a) :=
+    deterministic f hf a = Measure.dirac (f a) :=
   rfl
 #align probability_theory.kernel.deterministic_apply ProbabilityTheory.kernel.deterministic_apply
 
 theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : Set β}
-    (hs : MeasurableSet s) : deterministic hf a s = s.indicator (fun _ => 1) (f a) :=
+    (hs : MeasurableSet s) : deterministic f hf a s = s.indicator (fun _ => 1) (f a) :=
   by
   rw [deterministic]
   change measure.dirac (f a) s = s.indicator 1 (f a)
@@ -384,7 +384,7 @@ theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : S
 #align probability_theory.kernel.deterministic_apply' ProbabilityTheory.kernel.deterministic_apply'
 
 instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
-    IsMarkovKernel (deterministic hf) :=
+    IsMarkovKernel (deterministic f hf) :=
   ⟨fun a => by
     rw [deterministic_apply hf]
     infer_instance⟩

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

@@ -153,11 +153,11 @@ theorem kernel.measure_le_bound (κ : kernel α β) [h : IsFiniteKernel κ] (a :
   (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
 #align probability_theory.kernel.measure_le_bound ProbabilityTheory.kernel.measure_le_bound
 
-instance isFiniteKernelZero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
+instance isFiniteKernel_zero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
     IsFiniteKernel (0 : kernel α β) :=
   ⟨⟨0, ENNReal.coe_lt_top, fun a => by
       simp only [kernel.zero_apply, measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
-#align probability_theory.is_finite_kernel_zero ProbabilityTheory.isFiniteKernelZero
+#align probability_theory.is_finite_kernel_zero ProbabilityTheory.isFiniteKernel_zero
 
 instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :
     IsFiniteKernel (κ + η) :=
@@ -172,10 +172,10 @@ instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFinite
 
 variable {κ : kernel α β}
 
-instance IsMarkovKernel.isProbabilityMeasure' [h : IsMarkovKernel κ] (a : α) :
+instance IsMarkovKernel.is_probability_measure' [h : IsMarkovKernel κ] (a : α) :
     IsProbabilityMeasure (κ a) :=
   IsMarkovKernel.isProbabilityMeasure a
-#align probability_theory.is_markov_kernel.is_probability_measure' ProbabilityTheory.IsMarkovKernel.isProbabilityMeasure'
+#align probability_theory.is_markov_kernel.is_probability_measure' ProbabilityTheory.IsMarkovKernel.is_probability_measure'
 
 instance IsFiniteKernel.isFiniteMeasure [h : IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
   ⟨(kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
@@ -303,10 +303,10 @@ theorem measure_sum_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (a : α) :
     (Measure.sum fun n => seq κ n a) = κ a := by rw [← kernel.sum_apply, kernel_sum_seq κ]
 #align probability_theory.kernel.measure_sum_seq ProbabilityTheory.kernel.measure_sum_seq
 
-instance isFiniteKernelSeq (κ : kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :
+instance isFiniteKernel_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :
     IsFiniteKernel (kernel.seq κ n) :=
   h.tsum_finite.choose_spec.1 n
-#align probability_theory.kernel.is_finite_kernel_seq ProbabilityTheory.kernel.isFiniteKernelSeq
+#align probability_theory.kernel.is_finite_kernel_seq ProbabilityTheory.kernel.isFiniteKernel_seq
 
 instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
     IsSFiniteKernel (κ + η) :=
@@ -315,7 +315,7 @@ instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
   rw [sum_add, kernel_sum_seq κ, kernel_sum_seq η]
 #align probability_theory.kernel.is_s_finite_kernel.add ProbabilityTheory.kernel.IsSFiniteKernel.add
 
-theorem IsSFiniteKernel.finsetSum {κs : ι → kernel α β} (I : Finset ι)
+theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
     (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i in I, κs i) := by
   classical
     induction' I using Finset.induction with i I hi_nmem_I h_ind h
@@ -326,10 +326,10 @@ theorem IsSFiniteKernel.finsetSum {κs : ι → kernel α β} (I : Finset ι)
       have : is_s_finite_kernel (∑ x : ι in I, κs x) :=
         h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
       exact is_s_finite_kernel.add _ _
-#align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finsetSum
+#align probability_theory.kernel.is_s_finite_kernel.finset_sum ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i m) -/
-theorem isSFiniteKernelSumOfDenumerable [Denumerable ι] {κs : ι → kernel α β}
+theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel α β}
     (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (kernel.sum κs) :=
   by
   let e : ℕ ≃ ι × ℕ := Denumerable.equiv₂ ℕ (ι × ℕ)
@@ -343,10 +343,10 @@ theorem isSFiniteKernelSumOfDenumerable [Denumerable ι] {κs : ι → kernel α
   rw [e.tsum_eq]
   · rw [tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
   · infer_instance
-#align probability_theory.kernel.is_s_finite_kernel_sum_of_denumerable ProbabilityTheory.kernel.isSFiniteKernelSumOfDenumerable
+#align probability_theory.kernel.is_s_finite_kernel_sum_of_denumerable ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable
 
-theorem isSFiniteKernelSum [Countable ι] {κs : ι → kernel α β} (hκs : ∀ n, IsSFiniteKernel (κs n)) :
-    IsSFiniteKernel (kernel.sum κs) :=
+theorem isSFiniteKernel_sum [Countable ι] {κs : ι → kernel α β}
+    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (kernel.sum κs) :=
   by
   cases fintypeOrInfinite ι
   · rw [sum_fintype]
@@ -354,7 +354,7 @@ theorem isSFiniteKernelSum [Countable ι] {κs : ι → kernel α β} (hκs : 
   haveI : Encodable ι := Encodable.ofCountable ι
   haveI : Denumerable ι := Denumerable.ofEncodableOfInfinite ι
   exact is_s_finite_kernel_sum_of_denumerable hκs
-#align probability_theory.kernel.is_s_finite_kernel_sum ProbabilityTheory.kernel.isSFiniteKernelSum
+#align probability_theory.kernel.is_s_finite_kernel_sum ProbabilityTheory.kernel.isSFiniteKernel_sum
 
 end SFinite
 
@@ -383,12 +383,12 @@ theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : S
   simp_rw [measure.dirac_apply' _ hs]
 #align probability_theory.kernel.deterministic_apply' ProbabilityTheory.kernel.deterministic_apply'
 
-instance isMarkovKernelDeterministic {f : α → β} (hf : Measurable f) :
+instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
     IsMarkovKernel (deterministic hf) :=
   ⟨fun a => by
     rw [deterministic_apply hf]
     infer_instance⟩
-#align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernelDeterministic
+#align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
 
 end Deterministic
 
@@ -409,15 +409,15 @@ theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
   rfl
 #align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
 
-instance isFiniteKernelConst {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
+instance isFiniteKernel_const {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun a => le_rfl⟩⟩
-#align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernelConst
+#align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernel_const
 
-instance isMarkovKernelConst {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
+instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
     IsMarkovKernel (const α μβ) :=
   ⟨fun a => hμβ⟩
-#align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernelConst
+#align probability_theory.kernel.is_markov_kernel_const ProbabilityTheory.kernel.isMarkovKernel_const
 
 end Const
 
@@ -692,7 +692,7 @@ theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
 
 theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
     (f : α → β → ℝ≥0∞) :
-    @withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernelSum hκ) f =
+    @withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
       kernel.sum fun i => withDensity (κ i) f :=
   by
   by_cases hf : Measurable (Function.uncurry f)
@@ -724,14 +724,14 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
     by
     congr with b
     rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
-  rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).AeMeasurable]
+  rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).AEMeasurable]
   congr with n
   rw [with_density_apply' _ (hf n) a hs]
 #align probability_theory.kernel.with_density_tsum ProbabilityTheory.kernel.withDensity_tsum
 
 /-- If a kernel `κ` is finite and a function `f : α → β → ℝ≥0∞` is bounded, then `with_density κ f`
 is finite. -/
-theorem isFiniteKernelWithDensityOfBounded (κ : kernel α β) [IsFiniteKernel κ] {B : ℝ≥0∞}
+theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKernel κ] {B : ℝ≥0∞}
     (hB_top : B ≠ ∞) (hf_B : ∀ a b, f a b ≤ B) : IsFiniteKernel (withDensity κ f) :=
   by
   by_cases hf : Measurable (Function.uncurry f)
@@ -747,11 +747,11 @@ theorem isFiniteKernelWithDensityOfBounded (κ : kernel α β) [IsFiniteKernel 
             ⟩⟩
   · rw [with_density_of_not_measurable _ hf]
     infer_instance
-#align probability_theory.kernel.is_finite_kernel_with_density_of_bounded ProbabilityTheory.kernel.isFiniteKernelWithDensityOfBounded
+#align probability_theory.kernel.is_finite_kernel_with_density_of_bounded ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded
 
 /-- Auxiliary lemma for `is_s_finite_kernel_with_density`.
 If a kernel `κ` is finite, then `with_density κ f` is s-finite. -/
-theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFiniteKernel κ]
+theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFiniteKernel κ]
     (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) :=
   by
   -- We already have that for `f` bounded from above and a `κ` a finite kernel,
@@ -825,7 +825,7 @@ theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFinite
     _ ≤ n + 1 := (min_le_right _ _)
     _ = ↑(n + 1) := by norm_cast
     
-#align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernelWithDensityOfIsFiniteKernel
+#align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
 
 /-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
 `with_density κ f` is s-finite. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/da3fc4a33ff6bc75f077f691dc94c217b8d41559

@@ -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.basic
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
+! leanprover-community/mathlib commit 97d1aa955750bd57a7eeef91de310e633881670b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -405,6 +405,10 @@ def const (α : Type _) {β : Type _} [MeasurableSpace α] {mβ : MeasurableSpac
 
 include mα mβ
 
+theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
+  rfl
+#align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
+
 instance isFiniteKernelConst {μβ : Measure β} [hμβ : IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun a => le_rfl⟩⟩

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

@@ -818,7 +818,7 @@ theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFinite
   norm_cast
   calc
     fs n a b ≤ min (f a b) (n + 1) := tsub_le_self
-    _ ≤ n + 1 := min_le_right _ _
+    _ ≤ n + 1 := (min_le_right _ _)
     _ = ↑(n + 1) := by norm_cast
     
 #align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernelWithDensityOfIsFiniteKernel

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.basic
-! leanprover-community/mathlib commit 31a8a27684ce9a5749914f4248c3f7bf76605d41
+! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -739,8 +739,7 @@ theorem isFiniteKernelWithDensityOfBounded (κ : kernel α β) [IsFiniteKernel 
           calc
             (∫⁻ b in Set.univ, f a b ∂κ a) ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
             _ = B * κ a Set.univ := by simp only [measure.restrict_univ, lintegral_const]
-            _ ≤ B * is_finite_kernel.bound κ :=
-              ENNReal.mul_le_mul le_rfl (measure_le_bound κ a Set.univ)
+            _ ≤ B * is_finite_kernel.bound κ := mul_le_mul_left' (measure_le_bound κ a Set.univ) _
             ⟩⟩
   · rw [with_density_of_not_measurable _ hf]
     infer_instance

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

@@ -57,7 +57,7 @@ Particular kernels:
 
 open MeasureTheory
 
-open MeasureTheory Ennreal NNReal BigOperators
+open MeasureTheory ENNReal NNReal BigOperators
 
 namespace ProbabilityTheory
 
@@ -155,7 +155,7 @@ theorem kernel.measure_le_bound (κ : kernel α β) [h : IsFiniteKernel κ] (a :
 
 instance isFiniteKernelZero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
     IsFiniteKernel (0 : kernel α β) :=
-  ⟨⟨0, Ennreal.coe_lt_top, fun a => by
+  ⟨⟨0, ENNReal.coe_lt_top, fun a => by
       simp only [kernel.zero_apply, measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
 #align probability_theory.is_finite_kernel_zero ProbabilityTheory.isFiniteKernelZero
 
@@ -183,7 +183,7 @@ instance IsFiniteKernel.isFiniteMeasure [h : IsFiniteKernel κ] (a : α) : IsFin
 
 instance (priority := 100) IsMarkovKernel.isFiniteKernel [h : IsMarkovKernel κ] :
     IsFiniteKernel κ :=
-  ⟨⟨1, Ennreal.one_lt_top, fun a => prob_le_one⟩⟩
+  ⟨⟨1, ENNReal.one_lt_top, fun a => prob_le_one⟩⟩
 #align probability_theory.is_markov_kernel.is_finite_kernel ProbabilityTheory.IsMarkovKernel.isFiniteKernel
 
 namespace Kernel
@@ -222,7 +222,7 @@ protected noncomputable def sum [Countable ι] (κ : ι → kernel α β) : kern
   property := by
     refine' measure.measurable_of_measurable_coe _ fun s hs => _
     simp_rw [measure.sum_apply _ hs]
-    exact Measurable.ennreal_tsum fun n => kernel.measurable_coe (κ n) hs
+    exact Measurable.eNNReal_tsum fun n => kernel.measurable_coe (κ n) hs
 #align probability_theory.kernel.sum ProbabilityTheory.kernel.sum
 
 theorem sum_apply [Countable ι] (κ : ι → kernel α β) (a : α) :
@@ -262,7 +262,7 @@ theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
   by
   ext (a s hs)
   simp only [coe_fn_add, Pi.add_apply, sum_apply, measure.sum_apply _ hs, Pi.add_apply,
-    measure.coe_add, tsum_add Ennreal.summable Ennreal.summable]
+    measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
 #align probability_theory.kernel.sum_add ProbabilityTheory.kernel.sum_add
 
 end Sum
@@ -341,7 +341,7 @@ theorem isSFiniteKernelSumOfDenumerable [Denumerable ι] {κs : ι → kernel α
   simp_rw [kernel.sum_apply' _ _ hs]
   change (∑' (i) (m), seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
   rw [e.tsum_eq]
-  · rw [tsum_prod' Ennreal.summable fun _ => Ennreal.summable]
+  · rw [tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
   · infer_instance
 #align probability_theory.kernel.is_s_finite_kernel_sum_of_denumerable ProbabilityTheory.kernel.isSFiniteKernelSumOfDenumerable
 
@@ -549,7 +549,7 @@ theorem measurable_prod_mk_mem_of_finite (κ : kernel α β) {t : Set (α × β)
           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_prod_mk_mem_of_finite ProbabilityTheory.kernel.measurable_prod_mk_mem_of_finite
 
 theorem measurable_prod_mk_mem (κ : kernel α β) [IsSFiniteKernel κ] {t : Set (α × β)}
@@ -560,7 +560,7 @@ theorem measurable_prod_mk_mem (κ : kernel α β) [IsSFiniteKernel κ] {t : Set
     ∀ 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)
   simp_rw [this]
-  refine' Measurable.ennreal_tsum fun n => _
+  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
 
@@ -703,7 +703,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : ∀ i, Measurable (Function.uncurry (f i))) :
     withDensity κ (∑' n, f n) = kernel.sum fun n => withDensity κ (f n) :=
   by
-  have h_sum_a : ∀ a, Summable fun n => f n a := fun a => pi.summable.mpr fun b => Ennreal.summable
+  have h_sum_a : ∀ a, Summable fun n => f n a := fun a => pi.summable.mpr fun b => ENNReal.summable
   have h_sum : Summable fun n => f n := pi.summable.mpr h_sum_a
   ext (a s hs) : 2
   rw [sum_apply' _ a hs, with_density_apply' κ _ a hs]
@@ -713,9 +713,9 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
       ext1 p
       simp only [Function.uncurry_def]
       rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]
-      exact pi.summable.mpr fun p => Ennreal.summable
+      exact pi.summable.mpr fun p => ENNReal.summable
     rw [this]
-    exact Measurable.ennreal_tsum' hf
+    exact Measurable.eNNReal_tsum' hf
   have : (∫⁻ b in s, (∑' n, f n) a b ∂κ a) = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a :=
     by
     congr with b
@@ -733,14 +733,14 @@ theorem isFiniteKernelWithDensityOfBounded (κ : kernel α β) [IsFiniteKernel 
   by_cases hf : Measurable (Function.uncurry f)
   ·
     exact
-      ⟨⟨B * is_finite_kernel.bound κ, Ennreal.mul_lt_top hB_top (is_finite_kernel.bound_ne_top κ),
+      ⟨⟨B * is_finite_kernel.bound κ, ENNReal.mul_lt_top hB_top (is_finite_kernel.bound_ne_top κ),
           fun a => by
           rw [with_density_apply' κ hf a MeasurableSet.univ]
           calc
             (∫⁻ b in Set.univ, f a b ∂κ a) ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
             _ = B * κ a Set.univ := by simp only [measure.restrict_univ, lintegral_const]
             _ ≤ B * is_finite_kernel.bound κ :=
-              Ennreal.mul_le_mul le_rfl (measure_le_bound κ a Set.univ)
+              ENNReal.mul_le_mul le_rfl (measure_le_bound κ a Set.univ)
             ⟩⟩
   · rw [with_density_of_not_measurable _ hf]
     infer_instance
@@ -764,9 +764,9 @@ theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFinite
     by
     intro a b n hn
     have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
-    rw [← Ennreal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
+    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
     · refine' this.trans (le_of_eq _)
-      rw [Ennreal.ofReal_coe_nat]
+      rw [ENNReal.ofReal_coe_nat]
     · norm_cast
       exact zero_le _
   have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 :=
@@ -789,7 +789,7 @@ theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFinite
       exact h_zero a b n hn_not_mem
     ext (a b) : 2
     rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
-      Ennreal.tsum_eq_liminf_sum_nat]
+      ENNReal.tsum_eq_liminf_sum_nat]
     have h_finset_sum : ∀ n, (∑ i in Finset.range n, fs i a b) = min (f a b) n :=
       by
       intro n
@@ -815,7 +815,7 @@ theorem isSFiniteKernelWithDensityOfIsFiniteKernel (κ : kernel α β) [IsFinite
     by
     haveI := this
     infer_instance
-  refine' is_finite_kernel_with_density_of_bounded _ (Ennreal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => _
+  refine' is_finite_kernel_with_density_of_bounded _ (ENNReal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => _
   norm_cast
   calc
     fs n a b ≤ min (f a b) (n + 1) := tsub_le_self
@@ -843,7 +843,7 @@ theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel κ]
 /-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0`, `with_density κ f` is s-finite. -/
 instance (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0) :
     IsSFiniteKernel (withDensity κ fun a b => f a b) :=
-  IsSFiniteKernel.withDensity κ fun _ _ => Ennreal.coe_ne_top
+  IsSFiniteKernel.withDensity κ fun _ _ => ENNReal.coe_ne_top
 
 end WithDensity
 

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

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

@@ -296,7 +296,10 @@ class _root_.ProbabilityTheory.IsSFiniteKernel (κ : kernel α β) : Prop where
 
 instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
     IsSFiniteKernel κ :=
-  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by simp only; split_ifs; exact h; infer_instance, by
+  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by
+      simp only; split_ifs
+      · exact h
+      · infer_instance, by
       ext a s hs
       rw [kernel.sum_apply' _ _ hs]
       have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by

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

@@ -434,20 +434,28 @@ theorem integral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace 
   rw [kernel.deterministic_apply, integral_dirac _ (g a)]
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
 
-theorem set_integral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
-  rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
-#align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
+  rw [kernel.deterministic_apply, setIntegral_dirac' hf _ hs]
+#align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.setIntegral_deterministic'
+
+@[deprecated]
+alias set_integral_deterministic' :=
+  setIntegral_deterministic' -- deprecated on 2024-04-17
 
 @[simp]
-theorem set_integral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
-  rw [kernel.deterministic_apply, set_integral_dirac f _ s]
-#align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
+  rw [kernel.deterministic_apply, setIntegral_dirac f _ s]
+#align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.setIntegral_deterministic
+
+@[deprecated]
+alias set_integral_deterministic :=
+  setIntegral_deterministic -- deprecated on 2024-04-17
 
 end Deterministic
 
@@ -509,10 +517,14 @@ theorem integral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 
 @[simp]
-theorem set_integral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
     ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
-#align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
+#align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.setIntegral_const
+
+@[deprecated]
+alias set_integral_const :=
+  setIntegral_const -- deprecated on 2024-04-17
 
 end Const
 
@@ -565,11 +577,15 @@ theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : 
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
 
 @[simp]
-theorem set_integral_restrict {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_restrict {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
     ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, Measure.restrict_restrict' hs]
-#align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
+#align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.setIntegral_restrict
+
+@[deprecated]
+alias set_integral_restrict :=
+  setIntegral_restrict -- deprecated on 2024-04-17
 
 instance IsFiniteKernel.restrict (κ : kernel α β) [IsFiniteKernel κ] (hs : MeasurableSet s) :
     IsFiniteKernel (kernel.restrict κ hs) := by
@@ -713,12 +729,16 @@ theorem integral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
-theorem set_integral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem setIntegral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     (a : α) (g : β → E) (t : Set β) :
     ∫ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
-#align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise
+#align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.setIntegral_piecewise
+
+@[deprecated]
+alias set_integral_piecewise :=
+  setIntegral_piecewise -- deprecated on 2024-04-17
 
 end Piecewise
 

Let κ : kernel α (γ × β) and ν : kernel α γ be two finite kernels with kernel.fst κ ≤ ν, where γ has a countably generated σ-algebra (true in particular for standard Borel spaces). We build a function f : α → γ → Set β → ℝ jointly measurable in the first two arguments such that for all a : α and all measurable sets s : Set β and A : Set γ, ∫ x in A, f a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -169,6 +169,10 @@ instance IsFiniteKernel.add (κ η : kernel α β) [IsFiniteKernel κ] [IsFinite
   exact add_le_add (kernel.measure_le_bound _ _ _) (kernel.measure_le_bound _ _ _)
 #align probability_theory.is_finite_kernel.add ProbabilityTheory.IsFiniteKernel.add
 
+lemma isFiniteKernel_of_le {κ ν : kernel α β} [hν : IsFiniteKernel ν] (hκν : κ ≤ ν) :
+    IsFiniteKernel κ := by
+  refine ⟨hν.bound, hν.bound_lt_top, fun a ↦ (hκν _ _).trans (kernel.measure_le_bound ν a Set.univ)⟩
+
 variable {κ : kernel α β}
 
 instance IsMarkovKernel.is_probability_measure' [IsMarkovKernel κ] (a : α) :

Interaction of the composition-product of kernels with addition.

@@ -465,6 +465,9 @@ theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
 lemma const_zero : kernel.const α (0 : Measure β) = 0 := by
   ext x s _; simp [kernel.const_apply]
 
+lemma const_add (β : Type*) [MeasurableSpace β] (μ ν : Measure α) :
+    const β (μ + ν) = const β μ + const β ν := by ext; simp
+
 lemma sum_const [Countable ι] (μ : ι → Measure β) :
     kernel.sum (fun n ↦ const α (μ n)) = const α (Measure.sum μ) := by
   ext x s hs

Classifies by adding issue number #11215 to porting notes claiming "TODO".

@@ -133,8 +133,9 @@ class IsFiniteKernel (κ : kernel α β) : Prop where
 /-- A constant `C : ℝ≥0∞` such that `C < ∞` (`ProbabilityTheory.IsFiniteKernel.bound_lt_top κ`) and
 for all `a : α` and `s : Set β`, `κ a s ≤ C` (`ProbabilityTheory.kernel.measure_le_bound κ a s`).
 
-Porting note: TODO: does it make sense to make `ProbabilityTheory.IsFiniteKernel.bound` the least
-possible bound? Should it be an `NNReal` number? -/
+Porting note (#11215): TODO: does it make sense to
+-- make `ProbabilityTheory.IsFiniteKernel.bound` the least possible bound?
+-- Should it be an `NNReal` number? -/
 noncomputable def IsFiniteKernel.bound (κ : kernel α β) [h : IsFiniteKernel κ] : ℝ≥0∞ :=
   h.exists_univ_le.choose
 #align probability_theory.is_finite_kernel.bound ProbabilityTheory.IsFiniteKernel.bound

@@ -608,6 +608,10 @@ theorem comapRight_apply' (κ : kernel α β) (hf : MeasurableEmbedding f) (a :
     Measure.comap_apply _ hf.injective (fun s => hf.measurableSet_image.mpr) _ ht]
 #align probability_theory.kernel.comap_right_apply' ProbabilityTheory.kernel.comapRight_apply'
 
+@[simp]
+lemma comapRight_id (κ : kernel α β) : comapRight κ MeasurableEmbedding.id = κ := by
+  ext _ _ hs; rw [comapRight_apply' _ _ _ hs]; simp
+
 theorem IsMarkovKernel.comapRight (κ : kernel α β) (hf : MeasurableEmbedding f)
     (hκ : ∀ a, κ a (Set.range f) = 1) : IsMarkovKernel (comapRight κ hf) := by
   refine' ⟨fun a => ⟨_⟩⟩

@@ -70,6 +70,16 @@ instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
   coe := Subtype.val
   coe_injective' := Subtype.val_injective
 
+instance kernel.instCovariantAddLE {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
+    CovariantClass (kernel α β) (kernel α β) (· + ·) (· ≤ ·) :=
+  ⟨fun _ _ _ hμ a ↦ add_le_add_left (hμ a) _⟩
+
+noncomputable
+instance kernel.instOrderBot {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
+    OrderBot (kernel α β) where
+  bot := 0
+  bot_le κ a := by simp only [ZeroMemClass.coe_zero, Pi.zero_apply, Measure.zero_le]
+
 variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
 
 namespace kernel

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>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
 import Mathlib.MeasureTheory.Integral.Bochner
-import Mathlib.MeasureTheory.Constructions.Prod.Basic
+import Mathlib.MeasureTheory.Measure.GiryMonad
 
 #align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
 

This follows up from #9785, which renamed FunLike to DFunLike, by introducing a new abbreviation FunLike F α β := DFunLike F α (fun _ => β), to make the non-dependent use of FunLike easier.

I searched for the pattern DFunLike.*fun and DFunLike.*λ in all files to replace expressions of the form DFunLike F α (fun _ => β) with FunLike F α β. I did this everywhere except for extends clauses for two reasons: it would conflict with #8386, and more importantly extends must directly refer to a structure with no unfolding of defs or abbrevs.

@@ -64,9 +64,9 @@ noncomputable def kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace 
   add_mem' hf hg := Measurable.add hf hg
 #align probability_theory.kernel ProbabilityTheory.kernel
 
--- Porting note: using `DFunLike` instead of `CoeFun` to use `DFunLike.coe`
+-- Porting note: using `FunLike` instead of `CoeFun` to use `DFunLike.coe`
 instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
-    DFunLike (kernel α β) α fun _ => Measure β where
+    FunLike (kernel α β) α (Measure β) where
   coe := Subtype.val
   coe_injective' := Subtype.val_injective
 

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -64,9 +64,9 @@ noncomputable def kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace 
   add_mem' hf hg := Measurable.add hf hg
 #align probability_theory.kernel ProbabilityTheory.kernel
 
--- Porting note: using `FunLike` instead of `CoeFun` to use `FunLike.coe`
+-- Porting note: using `DFunLike` instead of `CoeFun` to use `DFunLike.coe`
 instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
-    FunLike (kernel α β) α fun _ => Measure β where
+    DFunLike (kernel α β) α fun _ => Measure β where
   coe := Subtype.val
   coe_injective' := Subtype.val_injective
 
@@ -177,10 +177,10 @@ instance (priority := 100) IsMarkovKernel.isFiniteKernel [IsMarkovKernel κ] :
 namespace kernel
 
 @[ext]
-theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η := FunLike.ext _ _ h
+theorem ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η := DFunLike.ext _ _ h
 #align probability_theory.kernel.ext ProbabilityTheory.kernel.ext
 
-theorem ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a := FunLike.ext_iff
+theorem ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a := DFunLike.ext_iff
 #align probability_theory.kernel.ext_iff ProbabilityTheory.kernel.ext_iff
 
 theorem ext_iff' {η : kernel α β} :

Subset of #9319

@@ -215,7 +215,7 @@ lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
     Integrable (fun x => (κ x s).toReal) μ := by
   refine' Integrable.mono' (integrable_const (IsFiniteKernel.bound κ).toReal)
     ((kernel.measurable_coe κ hs).ennreal_toReal.aestronglyMeasurable)
-    (ae_of_all μ <| fun x => _)
+    (ae_of_all μ fun x => _)
   rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg,
     ENNReal.toReal_le_toReal (measure_ne_top _ _) (IsFiniteKernel.bound_ne_top _)]
   exact kernel.measure_le_bound _ _ _

The composition-product of a measure and a kernel, denoted by ⊗ₘ, takes μ : Measure α and κ : kernel α β and creates μ ⊗ₘ κ : Measure (α × β). The integral of a function against μ ⊗ₘ κ is ∫⁻ x, f x ∂(μ ⊗ₘ κ) = ∫⁻ a, ∫⁻ b, f (a, b) ∂(κ a) ∂μ.

This PR also modifies the Disintegration file to use the new definition.

From the PFR project.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -454,11 +454,21 @@ theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
 lemma const_zero : kernel.const α (0 : Measure β) = 0 := by
   ext x s _; simp [kernel.const_apply]
 
+lemma sum_const [Countable ι] (μ : ι → Measure β) :
+    kernel.sum (fun n ↦ const α (μ n)) = const α (Measure.sum μ) := by
+  ext x s hs
+  rw [const_apply, Measure.sum_apply _ hs, kernel.sum_apply' _ _ hs]
+  simp only [const_apply]
+
 instance isFiniteKernel_const {μβ : Measure β} [IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun _ => le_rfl⟩⟩
 #align probability_theory.kernel.is_finite_kernel_const ProbabilityTheory.kernel.isFiniteKernel_const
 
+instance isSFiniteKernel_const {μβ : Measure β} [SFinite μβ] :
+    IsSFiniteKernel (const α μβ) :=
+  ⟨fun n ↦ const α (sFiniteSeq μβ n), fun n ↦ inferInstance, by rw [sum_const, sum_sFiniteSeq]⟩
+
 instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure μβ] :
     IsMarkovKernel (const α μβ) :=
   ⟨fun _ => hμβ⟩

@@ -97,7 +97,7 @@ theorem zero_apply (a : α) : (0 : kernel α β) a = 0 :=
 
 @[simp]
 theorem coe_finset_sum (I : Finset ι) (κ : ι → kernel α β) : ⇑(∑ i in I, κ i) = ∑ i in I, ⇑(κ i) :=
-  (coeAddHom α β).map_sum _ _
+  map_sum (coeAddHom α β) _ _
 #align probability_theory.kernel.coe_finset_sum ProbabilityTheory.kernel.coe_finset_sum
 
 theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :

@@ -445,10 +445,15 @@ def const (α : Type*) {β : Type*} [MeasurableSpace α] {_ : MeasurableSpace β
   property := measurable_const
 #align probability_theory.kernel.const ProbabilityTheory.kernel.const
 
+@[simp]
 theorem const_apply (μβ : Measure β) (a : α) : const α μβ a = μβ :=
   rfl
 #align probability_theory.kernel.const_apply ProbabilityTheory.kernel.const_apply
 
+@[simp]
+lemma const_zero : kernel.const α (0 : Measure β) = 0 := by
+  ext x s _; simp [kernel.const_apply]
+
 instance isFiniteKernel_const {μβ : Measure β} [IsFiniteMeasure μβ] :
     IsFiniteKernel (const α μβ) :=
   ⟨⟨μβ Set.univ, measure_lt_top _ _, fun _ => le_rfl⟩⟩

@sgouezel added a version of withDensity_apply that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about rnDeriv, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: RemyDegenne <Remydegenne@gmail.com>

@@ -308,6 +308,9 @@ instance isFiniteKernel_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (n : 
   h.tsum_finite.choose_spec.1 n
 #align probability_theory.kernel.is_finite_kernel_seq ProbabilityTheory.kernel.isFiniteKernel_seq
 
+instance IsSFiniteKernel.sFinite [IsSFiniteKernel κ] (a : α) : SFinite (κ a) :=
+  ⟨⟨fun n ↦ seq κ n a, inferInstance, (measure_sum_seq κ a).symm⟩⟩
+
 instance IsSFiniteKernel.add (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
     IsSFiniteKernel (κ + η) := by
   refine' ⟨⟨fun n => seq κ n + seq η n, fun n => inferInstance, _⟩⟩

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

This has nice performance benefits.

@@ -57,7 +57,7 @@ namespace ProbabilityTheory
 `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
 `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
 iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`. -/
-noncomputable def kernel (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :
+noncomputable def kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
     AddSubmonoid (α → Measure β) where
   carrier := Measurable
   zero_mem' := measurable_zero
@@ -65,12 +65,12 @@ noncomputable def kernel (α β : Type _) [MeasurableSpace α] [MeasurableSpace
 #align probability_theory.kernel ProbabilityTheory.kernel
 
 -- Porting note: using `FunLike` instead of `CoeFun` to use `FunLike.coe`
-instance {α β : Type _} [MeasurableSpace α] [MeasurableSpace β] :
+instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
     FunLike (kernel α β) α fun _ => Measure β where
   coe := Subtype.val
   coe_injective' := Subtype.val_injective
 
-variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
 
 namespace kernel
 
@@ -85,7 +85,7 @@ theorem coeFn_add (κ η : kernel α β) : ⇑(κ + η) = κ + η :=
 #align probability_theory.kernel.coe_fn_add ProbabilityTheory.kernel.coeFn_add
 
 /-- Coercion to a function as an additive monoid homomorphism. -/
-def coeAddHom (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :
+def coeAddHom (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
     kernel α β →+ α → Measure β :=
   AddSubmonoid.subtype _
 #align probability_theory.kernel.coe_add_hom ProbabilityTheory.kernel.coeAddHom
@@ -144,7 +144,7 @@ theorem kernel.measure_le_bound (κ : kernel α β) [h : IsFiniteKernel κ] (a :
   (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
 #align probability_theory.kernel.measure_le_bound ProbabilityTheory.kernel.measure_le_bound
 
-instance isFiniteKernel_zero (α β : Type _) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
+instance isFiniteKernel_zero (α β : Type*) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
     IsFiniteKernel (0 : kernel α β) :=
   ⟨⟨0, ENNReal.coe_lt_top, fun _ => by
       simp only [kernel.zero_apply, Measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
@@ -403,20 +403,20 @@ theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a :
   rw [kernel.deterministic_apply, set_lintegral_dirac f s]
 #align probability_theory.kernel.set_lintegral_deterministic ProbabilityTheory.kernel.set_lintegral_deterministic
 
-theorem integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem integral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac' _ _ hf]
 #align probability_theory.kernel.integral_deterministic' ProbabilityTheory.kernel.integral_deterministic'
 
 @[simp]
-theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem integral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac _ (g a)]
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
 
-theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem set_integral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
@@ -424,7 +424,7 @@ theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedS
 #align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
 
 @[simp]
-theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem set_integral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
     ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
@@ -436,7 +436,7 @@ end Deterministic
 section Const
 
 /-- Constant kernel, which always returns the same measure. -/
-def const (α : Type _) {β : Type _} [MeasurableSpace α] {_ : MeasurableSpace β} (μβ : Measure β) :
+def const (α : Type*) {β : Type*} [MeasurableSpace α] {_ : MeasurableSpace β} (μβ : Measure β) :
     kernel α β where
   val _ := μβ
   property := measurable_const
@@ -467,13 +467,13 @@ theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {
 #align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
 
 @[simp]
-theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem integral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {μ : Measure β} {a : α} : ∫ x, f x ∂kernel.const α μ a = ∫ x, f x ∂μ := by
   rw [kernel.const_apply]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 
 @[simp]
-theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem set_integral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
     ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
@@ -529,7 +529,7 @@ theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : 
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
 
 @[simp]
-theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem set_integral_restrict {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
     ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, Measure.restrict_restrict' hs]
@@ -553,7 +553,7 @@ end Restrict
 
 section ComapRight
 
-variable {γ : Type _} {mγ : MeasurableSpace γ} {f : γ → β}
+variable {γ : Type*} {mγ : MeasurableSpace γ} {f : γ → β}
 
 /-- Kernel with value `(κ a).comap f`, for a measurable embedding `f`. That is, for a measurable set
 `t : Set β`, `ProbabilityTheory.kernel.comapRight κ hf a t = κ a (f '' t)`. -/
@@ -667,13 +667,13 @@ theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
 
-theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem integral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     (a : α) (g : β → E) :
     ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
-theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem set_integral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     (a : α) (g : β → E) (t : Set β) :
     ∫ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=

Co-authored-by: JasonKYi <kexing.ying19@imperial.ac.uk>

@@ -210,6 +210,21 @@ protected theorem measurable_coe (κ : kernel α β) {s : Set β} (hs : Measurab
   (Measure.measurable_coe hs).comp (kernel.measurable κ)
 #align probability_theory.kernel.measurable_coe ProbabilityTheory.kernel.measurable_coe
 
+lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
+    (κ : kernel α β) [IsFiniteKernel κ] {s : Set β} (hs : MeasurableSet s) :
+    Integrable (fun x => (κ x s).toReal) μ := by
+  refine' Integrable.mono' (integrable_const (IsFiniteKernel.bound κ).toReal)
+    ((kernel.measurable_coe κ hs).ennreal_toReal.aestronglyMeasurable)
+    (ae_of_all μ <| fun x => _)
+  rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg,
+    ENNReal.toReal_le_toReal (measure_ne_top _ _) (IsFiniteKernel.bound_ne_top _)]
+  exact kernel.measure_le_bound _ _ _
+
+lemma IsMarkovKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
+    (κ : kernel α β) [IsMarkovKernel κ] {s : Set β} (hs : MeasurableSet s) :
+    Integrable (fun x => (κ x s).toReal) μ :=
+  IsFiniteKernel.integrable μ κ hs
+
 section Sum
 
 /-- Sum of an indexed family of kernels. -/

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 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.basic
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Integral.Bochner
 import Mathlib.MeasureTheory.Constructions.Prod.Basic
 
+#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
+
 /-!
 # Markov Kernels
 

The notion of Bochner integral of a function taking values in a normed space E requires that E is complete. This means that whenever we write down an integral, the term contains the assertion that E is complete.

In this PR, we remove the completeness requirement from the definition, using the junk value 0 when the space is not complete. Mathematically this does not make any difference, as all reasonable applications will be with a complete E. But it means that terms involving integrals become a little bit simpler and that completeness will not have to be checked by the system when applying a bunch of basic lemmas on integrals.

@@ -455,13 +455,13 @@ theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {
 #align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
 
 @[simp]
-theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {μ : Measure β} {a : α} : ∫ x, f x ∂kernel.const α μ a = ∫ x, f x ∂μ := by
   rw [kernel.const_apply]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 
 @[simp]
-theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
     ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
@@ -518,7 +518,7 @@ theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : 
 
 @[simp]
 theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
+    {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
     ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, Measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
@@ -655,14 +655,14 @@ theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
 
-theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     (a : α) (g : β → E) :
     ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
 theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
+    (a : α) (g : β → E) (t : Set β) :
     ∫ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl

@@ -191,7 +191,7 @@ theorem ext_iff' {η : kernel α β} :
   simp_rw [ext_iff, Measure.ext_iff]
 #align probability_theory.kernel.ext_iff' ProbabilityTheory.kernel.ext_iff'
 
-theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a) :
+theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
     κ = η := by
   ext a s hs
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
@@ -200,7 +200,7 @@ theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 
 theorem ext_fun_iff {η : kernel α β} :
-    κ = η ↔ ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a :=
+    κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a :=
   ⟨fun h a f _ => by rw [h], ext_fun⟩
 #align probability_theory.kernel.ext_fun_iff ProbabilityTheory.kernel.ext_fun_iff
 
@@ -368,46 +368,46 @@ instance isMarkovKernel_deterministic {f : α → β} (hf : Measurable f) :
 #align probability_theory.kernel.is_markov_kernel_deterministic ProbabilityTheory.kernel.isMarkovKernel_deterministic
 
 theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : Measurable f) : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    (hf : Measurable f) : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac' _ hf]
 #align probability_theory.kernel.lintegral_deterministic' ProbabilityTheory.kernel.lintegral_deterministic'
 
 @[simp]
 theorem lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] : (∫⁻ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    [MeasurableSingletonClass β] : ∫⁻ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, lintegral_dirac (g a) f]
 #align probability_theory.kernel.lintegral_deterministic ProbabilityTheory.kernel.lintegral_deterministic
 
 theorem set_lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
-    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]
 #align probability_theory.kernel.set_lintegral_deterministic' ProbabilityTheory.kernel.set_lintegral_deterministic'
 
 @[simp]
 theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
-    (∫⁻ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_lintegral_dirac f s]
 #align probability_theory.kernel.set_lintegral_deterministic ProbabilityTheory.kernel.set_lintegral_deterministic
 
 theorem integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : StronglyMeasurable f) : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    (hf : StronglyMeasurable f) : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac' _ _ hf]
 #align probability_theory.kernel.integral_deterministic' ProbabilityTheory.kernel.integral_deterministic'
 
 @[simp]
 theorem integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] : (∫ x, f x ∂kernel.deterministic g hg a) = f (g a) := by
+    [MeasurableSingletonClass β] : ∫ x, f x ∂kernel.deterministic g hg a = f (g a) := by
   rw [kernel.deterministic_apply, integral_dirac _ (g a)]
 #align probability_theory.kernel.integral_deterministic ProbabilityTheory.kernel.integral_deterministic
 
 theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
-    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]
 #align probability_theory.kernel.set_integral_deterministic' ProbabilityTheory.kernel.set_integral_deterministic'
 
@@ -415,7 +415,7 @@ theorem set_integral_deterministic' {E : Type _} [NormedAddCommGroup E] [NormedS
 theorem set_integral_deterministic {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
     [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
-    (∫ x in s, f x ∂kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 := by
+    ∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
   rw [kernel.deterministic_apply, set_integral_dirac f _ s]
 #align probability_theory.kernel.set_integral_deterministic ProbabilityTheory.kernel.set_integral_deterministic
 
@@ -446,24 +446,24 @@ instance isMarkovKernel_const {μβ : Measure β} [hμβ : IsProbabilityMeasure
 
 @[simp]
 theorem lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} :
-    (∫⁻ x, f x ∂kernel.const α μ a) = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
+    ∫⁻ x, f x ∂kernel.const α μ a = ∫⁻ x, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.lintegral_const ProbabilityTheory.kernel.lintegral_const
 
 @[simp]
 theorem set_lintegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {s : Set β} :
-    (∫⁻ x in s, f x ∂kernel.const α μ a) = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
+    ∫⁻ x in s, f x ∂kernel.const α μ a = ∫⁻ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_lintegral_const ProbabilityTheory.kernel.set_lintegral_const
 
 @[simp]
 theorem integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-    {f : β → E} {μ : Measure β} {a : α} : (∫ x, f x ∂kernel.const α μ a) = ∫ x, f x ∂μ := by
+    {f : β → E} {μ : Measure β} {a : α} : ∫ x, f x ∂kernel.const α μ a = ∫ x, f x ∂μ := by
   rw [kernel.const_apply]
 #align probability_theory.kernel.integral_const ProbabilityTheory.kernel.integral_const
 
 @[simp]
 theorem set_integral_const {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
-    (∫ x in s, f x ∂kernel.const α μ a) = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
+    ∫ x in s, f x ∂kernel.const α μ a = ∫ x in s, f x ∂μ := by rw [kernel.const_apply]
 #align probability_theory.kernel.set_integral_const ProbabilityTheory.kernel.set_integral_const
 
 end Const
@@ -507,19 +507,19 @@ theorem restrict_univ : kernel.restrict κ MeasurableSet.univ = κ := by
 
 @[simp]
 theorem lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞) :
-    (∫⁻ b, f b ∂kernel.restrict κ hs a) = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
+    ∫⁻ b, f b ∂kernel.restrict κ hs a = ∫⁻ b in s, f b ∂κ a := by rw [restrict_apply]
 #align probability_theory.kernel.lintegral_restrict ProbabilityTheory.kernel.lintegral_restrict
 
 @[simp]
 theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞)
-    (t : Set β) : (∫⁻ b in t, f b ∂kernel.restrict κ hs a) = ∫⁻ b in t ∩ s, f b ∂κ a := by
+    (t : Set β) : ∫⁻ b in t, f b ∂kernel.restrict κ hs a = ∫⁻ b in t ∩ s, f b ∂κ a := by
   rw [restrict_apply, Measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_lintegral_restrict ProbabilityTheory.kernel.set_lintegral_restrict
 
 @[simp]
 theorem set_integral_restrict {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
-    (∫ x in t, f x ∂kernel.restrict κ hs a) = ∫ x in t ∩ s, f x ∂κ a := by
+    ∫ x in t, f x ∂kernel.restrict κ hs a = ∫ x in t ∩ s, f x ∂κ a := by
   rw [restrict_apply, Measure.restrict_restrict' hs]
 #align probability_theory.kernel.set_integral_restrict ProbabilityTheory.kernel.set_integral_restrict
 
@@ -645,25 +645,25 @@ protected instance IsSFiniteKernel.piecewise [IsSFiniteKernel κ] [IsSFiniteKern
 #align probability_theory.kernel.is_s_finite_kernel.piecewise ProbabilityTheory.kernel.IsSFiniteKernel.piecewise
 
 theorem lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
-    (∫⁻ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
+    ∫⁻ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫⁻ b, g b ∂κ a else ∫⁻ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.lintegral_piecewise ProbabilityTheory.kernel.lintegral_piecewise
 
 theorem set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
-    (∫⁻ b in t, g b ∂piecewise hs κ η a) =
+    ∫⁻ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫⁻ b in t, g b ∂κ a else ∫⁻ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_lintegral_piecewise ProbabilityTheory.kernel.set_lintegral_piecewise
 
 theorem integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     (a : α) (g : β → E) :
-    (∫ b, g b ∂piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
+    ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
   simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.integral_piecewise ProbabilityTheory.kernel.integral_piecewise
 
 theorem set_integral_piecewise {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (a : α) (g : β → E) (t : Set β) :
-    (∫ b in t, g b ∂piecewise hs κ η a) =
+    ∫ b in t, g b ∂piecewise hs κ η a =
       if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a :=
   by simp_rw [piecewise_apply]; split_ifs <;> rfl
 #align probability_theory.kernel.set_integral_piecewise ProbabilityTheory.kernel.set_integral_piecewise

@@ -23,7 +23,7 @@ measurable sets `s` of `β`, `a ↦ κ a s` is measurable.
 
 Classes of kernels:
 * `ProbabilityTheory.kernel α β`: kernels from `α` to `β`, defined as the `AddSubmonoid` of the
-  measurable functions in `α → measure β`.
+  measurable functions in `α → Measure β`.
 * `ProbabilityTheory.IsMarkovKernel κ`: a kernel from `α` to `β` is said to be a Markov kernel
   if for all `a : α`, `k a` is a probability measure.
 * `ProbabilityTheory.IsFiniteKernel κ`: a kernel from `α` to `β` is said to be finite if there
@@ -36,7 +36,7 @@ Classes of kernels:
 
 Particular kernels:
 * `ProbabilityTheory.kernel.deterministic (f : α → β) (hf : Measurable f)`:
-  kernel `a ↦ measure.dirac (f a)`.
+  kernel `a ↦ Measure.dirac (f a)`.
 * `ProbabilityTheory.kernel.const α (μβ : measure β)`: constant kernel `a ↦ μβ`.
 * `ProbabilityTheory.kernel.restrict κ (hs : MeasurableSet s)`: kernel for which the image of
   `a : α` is `(κ a).restrict s`.
@@ -57,7 +57,7 @@ open scoped MeasureTheory ENNReal NNReal BigOperators
 namespace ProbabilityTheory
 
 /-- A kernel from a measurable space `α` to another measurable space `β` is a measurable function
-`κ : α → measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
+`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
 `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
 iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`. -/
 noncomputable def kernel (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -193,10 +193,9 @@ theorem ext_iff' {η : kernel α β} :
 
 theorem ext_fun {η : kernel α β} (h : ∀ a f, Measurable f → (∫⁻ b, f b ∂κ a) = ∫⁻ b, f b ∂η a) :
     κ = η := by
-   -- porting note: why `ext` doesn't accept `hs` as a third argument (here and below)?
-  ext a s; intro hs
+  ext a s hs
   specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
-  simp_rw [lintegral_indicator_const hs, one_mul] at h 
+  simp_rw [lintegral_indicator_const hs, one_mul] at h
   rw [h]
 #align probability_theory.kernel.ext_fun ProbabilityTheory.kernel.ext_fun
 
@@ -236,25 +235,25 @@ theorem sum_apply' [Countable ι] (κ : ι → kernel α β) (a : α) {s : Set 
 
 @[simp]
 theorem sum_zero [Countable ι] : (kernel.sum fun _ : ι => (0 : kernel α β)) = 0 := by
-  ext a s; intro hs
+  ext a s hs
   rw [sum_apply' _ a hs]
   simp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero]
 #align probability_theory.kernel.sum_zero ProbabilityTheory.kernel.sum_zero
 
 theorem sum_comm [Countable ι] (κ : ι → ι → kernel α β) :
     (kernel.sum fun n => kernel.sum (κ n)) = kernel.sum fun m => kernel.sum fun n => κ n m := by
-  ext a s; intro; simp_rw [sum_apply]; rw [Measure.sum_comm]
+  ext a s; simp_rw [sum_apply]; rw [Measure.sum_comm]
 #align probability_theory.kernel.sum_comm ProbabilityTheory.kernel.sum_comm
 
 @[simp]
 theorem sum_fintype [Fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i := by
-  ext a s; intro hs
+  ext a s hs
   simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
 #align probability_theory.kernel.sum_fintype ProbabilityTheory.kernel.sum_fintype
 
 theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
     (kernel.sum fun n => κ n + η n) = kernel.sum κ + kernel.sum η := by
-  ext a s; intro hs
+  ext a s hs
   simp only [coeFn_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply,
     Measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
 #align probability_theory.kernel.sum_add ProbabilityTheory.kernel.sum_add
@@ -271,7 +270,7 @@ class _root_.ProbabilityTheory.IsSFiniteKernel (κ : kernel α β) : Prop where
 instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
     IsSFiniteKernel κ :=
   ⟨⟨fun n => if n = 0 then κ else 0, fun n => by simp only; split_ifs; exact h; infer_instance, by
-      ext a s; intro hs
+      ext a s hs
       rw [kernel.sum_apply' _ _ hs]
       have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by
         ext1 i; split_ifs <;> rfl
@@ -321,7 +320,7 @@ theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → kernel
   refine' ⟨⟨fun n => seq (κs (e n).1) (e n).2, inferInstance, _⟩⟩
   have hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) := by
     simp_rw [kernel_sum_seq]
-  ext a s; intro hs
+  ext a s hs
   rw [hκ_eq]
   simp_rw [kernel.sum_apply' _ _ hs]
   change (∑' i, ∑' m, seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
@@ -674,4 +673,3 @@ end Piecewise
 end kernel
 
 end ProbabilityTheory
-

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

@@ -59,7 +59,7 @@ namespace ProbabilityTheory
 /-- A kernel from a measurable space `α` to another measurable space `β` is a measurable function
 `κ : α → measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
 `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
-iff `∀ s : Set β, MeasurableSet s → Measurable (λ a, κ a s)`. -/
+iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`. -/
 noncomputable def kernel (α β : Type _) [MeasurableSpace α] [MeasurableSpace β] :
     AddSubmonoid (α → Measure β) where
   carrier := Measurable

Part 3 of #5001

@@ -28,7 +28,7 @@ Classes of kernels:
   if for all `a : α`, `k a` is a probability measure.
 * `ProbabilityTheory.IsFiniteKernel κ`: a kernel from `α` to `β` is said to be finite if there
   exists `C : ℝ≥0∞` such that `C < ∞` and for all `a : α`, `κ a univ ≤ C`. This implies in
-  particular that all measures in the image of `κ` are finite, but is stronger since it requires an
+  particular that all measures in the image of `κ` are finite, but is stronger since it requires a
   uniform bound. This stronger condition is necessary to ensure that the composition of two finite
   kernels is finite.
 * `ProbabilityTheory.IsSFiniteKernel κ`: a kernel is called s-finite if it is a countable