probability.kernel.cond_distrib ⟷ Mathlib.Probability.Kernel.CondDistrib

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

@@ -3,7 +3,7 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Probability.Kernel.Disintegration
+import Probability.Kernel.Disintegration.Basic
 import Probability.Notation
 
 #align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"

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

@@ -3,8 +3,8 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathbin.Probability.Kernel.Disintegration
-import Mathbin.Probability.Notation
+import Probability.Kernel.Disintegration
+import Probability.Notation
 
 #align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.cond_distrib
-! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Kernel.Disintegration
 import Mathbin.Probability.Notation
 
+#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
+
 /-!
 # Regular conditional probability distribution
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

@@ -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.cond_distrib
-! leanprover-community/mathlib commit 00abe0695d8767201e6d008afa22393978bb324d
+! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Probability.Notation
 /-!
 # Regular conditional probability distribution
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define the regular conditional probability distribution of `Y : α → Ω` given `X : α → β`, where
 `Ω` is a standard Borel space. This is a `kernel β Ω` such that for almost all `a`, `cond_distrib`
 evaluated at `X a` and a measurable set `s` is equal to the conditional expectation

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -56,6 +56,7 @@ variable {α β Ω F : Type _} [TopologicalSpace Ω] [MeasurableSpace Ω] [Polis
   [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ]
   {X : α → β} {Y : α → Ω}
 
+#print ProbabilityTheory.condDistrib /-
 /-- **Regular conditional probability distribution**: kernel associated with the conditional
 expectation of `Y` given `X`.
 For almost all `a`, `cond_distrib Y X μ` evaluated at `X a` and a measurable set `s` is equal to
@@ -66,6 +67,7 @@ noncomputable irreducible_def condDistrib {mα : MeasurableSpace α} [Measurable
     (X : α → β) (μ : Measure α) [IsFiniteMeasure μ] : kernel β Ω :=
   (μ.map fun a => (X a, Y a)).condKernel
 #align probability_theory.cond_distrib ProbabilityTheory.condDistrib
+-/
 
 instance [MeasurableSpace β] : IsMarkovKernel (condDistrib Y X μ) := by rw [cond_distrib];
   infer_instance
@@ -74,11 +76,14 @@ variable {mβ : MeasurableSpace β} {s : Set Ω} {t : Set β} {f : β × Ω →
 
 section Measurability
 
+#print ProbabilityTheory.measurable_condDistrib /-
 theorem measurable_condDistrib (hs : MeasurableSet s) :
     measurable[mβ.comap X] fun a => condDistrib Y X μ (X a) s :=
   (kernel.measurable_coe _ hs).comp (Measurable.of_comap_le le_rfl)
 #align probability_theory.measurable_cond_distrib ProbabilityTheory.measurable_condDistrib
+-/
 
+#print MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff /-
 theorem MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     (∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
@@ -86,31 +91,39 @@ theorem MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff (hX
       Integrable f (μ.map fun a => (X a, Y a)) :=
   by rw [cond_distrib, ← hf.ae_integrable_cond_kernel_iff, measure.fst_map_prod_mk₀ hX hY]
 #align measure_theory.ae_strongly_measurable.ae_integrable_cond_distrib_map_iff MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
+-/
 
 variable [NormedSpace ℝ F] [CompleteSpace F]
 
+#print MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map /-
 theorem MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) := by
   rw [← measure.fst_map_prod_mk₀ hX hY, cond_distrib]; exact hf.integral_cond_kernel
 #align measure_theory.ae_strongly_measurable.integral_cond_distrib_map MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
+-/
 
+#print MeasureTheory.AEStronglyMeasurable.integral_condDistrib /-
 theorem MeasureTheory.AEStronglyMeasurable.integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
   (hf.integral_condDistrib_map hX hY).comp_aemeasurable hX
 #align measure_theory.ae_strongly_measurable.integral_cond_distrib MeasureTheory.AEStronglyMeasurable.integral_condDistrib
+-/
 
-theorem aEStronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
+#print ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib /-
+theorem aestronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
     (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable' (mβ.comap X) (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
   (hf.integral_condDistrib_map hX hY).comp_ae_measurable' hX
-#align probability_theory.ae_strongly_measurable'_integral_cond_distrib ProbabilityTheory.aEStronglyMeasurable'_integral_condDistrib
+#align probability_theory.ae_strongly_measurable'_integral_cond_distrib ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib
+-/
 
 end Measurability
 
 section Integrability
 
+#print ProbabilityTheory.integrable_toReal_condDistrib /-
 theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :
     Integrable (fun a => (condDistrib Y X μ (X a) s).toReal) μ :=
   by
@@ -122,60 +135,78 @@ theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableS
       _ = μ univ := lintegral_one
       _ < ∞ := measure_lt_top _ _
 #align probability_theory.integrable_to_real_cond_distrib ProbabilityTheory.integrable_toReal_condDistrib
+-/
 
+#print MeasureTheory.Integrable.condDistrib_ae_map /-
 theorem MeasureTheory.Integrable.condDistrib_ae_map (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
     (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     ∀ᵐ b ∂μ.map X, Integrable (fun ω => f (b, ω)) (condDistrib Y X μ b) := by
   rw [cond_distrib, ← measure.fst_map_prod_mk₀ hX hY]; exact hf_int.cond_kernel_ae
 #align measure_theory.integrable.cond_distrib_ae_map MeasureTheory.Integrable.condDistrib_ae_map
+-/
 
+#print MeasureTheory.Integrable.condDistrib_ae /-
 theorem MeasureTheory.Integrable.condDistrib_ae (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
     (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     ∀ᵐ a ∂μ, Integrable (fun ω => f (X a, ω)) (condDistrib Y X μ (X a)) :=
   ae_of_ae_map hX (hf_int.condDistrib_ae_map hX hY)
 #align measure_theory.integrable.cond_distrib_ae MeasureTheory.Integrable.condDistrib_ae
+-/
 
+#print MeasureTheory.Integrable.integral_norm_condDistrib_map /-
 theorem MeasureTheory.Integrable.integral_norm_condDistrib_map (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) := by
   rw [cond_distrib, ← measure.fst_map_prod_mk₀ hX hY]; exact hf_int.integral_norm_cond_kernel
 #align measure_theory.integrable.integral_norm_cond_distrib_map MeasureTheory.Integrable.integral_norm_condDistrib_map
+-/
 
+#print MeasureTheory.Integrable.integral_norm_condDistrib /-
 theorem MeasureTheory.Integrable.integral_norm_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ∫ y, ‖f (X a, y)‖ ∂condDistrib Y X μ (X a)) μ :=
   (hf_int.integral_norm_condDistrib_map hX hY).comp_aemeasurable hX
 #align measure_theory.integrable.integral_norm_cond_distrib MeasureTheory.Integrable.integral_norm_condDistrib
+-/
 
 variable [NormedSpace ℝ F] [CompleteSpace F]
 
+#print MeasureTheory.Integrable.norm_integral_condDistrib_map /-
 theorem MeasureTheory.Integrable.norm_integral_condDistrib_map (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) := by
   rw [cond_distrib, ← measure.fst_map_prod_mk₀ hX hY]; exact hf_int.norm_integral_cond_kernel
 #align measure_theory.integrable.norm_integral_cond_distrib_map MeasureTheory.Integrable.norm_integral_condDistrib_map
+-/
 
+#print MeasureTheory.Integrable.norm_integral_condDistrib /-
 theorem MeasureTheory.Integrable.norm_integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ‖∫ y, f (X a, y) ∂condDistrib Y X μ (X a)‖) μ :=
   (hf_int.norm_integral_condDistrib_map hX hY).comp_aemeasurable hX
 #align measure_theory.integrable.norm_integral_cond_distrib MeasureTheory.Integrable.norm_integral_condDistrib
+-/
 
+#print MeasureTheory.Integrable.integral_condDistrib_map /-
 theorem MeasureTheory.Integrable.integral_condDistrib_map (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) :=
   (integrable_norm_iff (hf_int.1.integral_condDistrib_map hX hY)).mp
     (hf_int.norm_integral_condDistrib_map hX hY)
 #align measure_theory.integrable.integral_cond_distrib_map MeasureTheory.Integrable.integral_condDistrib_map
+-/
 
+#print MeasureTheory.Integrable.integral_condDistrib /-
 theorem MeasureTheory.Integrable.integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
   (hf_int.integral_condDistrib_map hX hY).comp_aemeasurable hX
 #align measure_theory.integrable.integral_cond_distrib MeasureTheory.Integrable.integral_condDistrib
+-/
 
 end Integrability
 
+#print ProbabilityTheory.set_lintegral_preimage_condDistrib /-
 theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hs : MeasurableSet s) (ht : MeasurableSet t) :
     ∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by
@@ -183,13 +214,17 @@ theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurabl
     measure.fst_map_prod_mk₀ hX.ae_measurable hY, set_lintegral_cond_kernel_eq_measure_prod _ ht hs,
     measure.map_apply_of_ae_measurable (hX.ae_measurable.prod_mk hY) (ht.prod hs), mk_preimage_prod]
 #align probability_theory.set_lintegral_preimage_cond_distrib ProbabilityTheory.set_lintegral_preimage_condDistrib
+-/
 
+#print ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet /-
 theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hs : MeasurableSet s) {t : Set α} (ht : measurable_set[mβ.comap X] t) :
     ∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by obtain ⟨t', ht', rfl⟩ := ht;
   rw [set_lintegral_preimage_cond_distrib hX hY hs ht']
 #align probability_theory.set_lintegral_cond_distrib_of_measurable_set ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet
+-/
 
+#print ProbabilityTheory.condDistrib_ae_eq_condexp /-
 /-- For almost every `a : α`, the `cond_distrib Y X μ` kernel applied to `X a` and a measurable set
 `s` is equal to the conditional expectation of the indicator of `Y ⁻¹' s`. -/
 theorem condDistrib_ae_eq_condexp (hX : Measurable X) (hY : Measurable Y) (hs : MeasurableSet s) :
@@ -206,10 +241,12 @@ theorem condDistrib_ae_eq_condexp (hX : Measurable X) (hY : Measurable Y) (hs :
   · refine' (Measurable.stronglyMeasurable _).aestronglyMeasurable'
     exact @Measurable.ennreal_toReal _ (mβ.comap X) _ (measurable_cond_distrib hs)
 #align probability_theory.cond_distrib_ae_eq_condexp ProbabilityTheory.condDistrib_ae_eq_condexp
+-/
 
+#print ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib' /-
 /-- The conditional expectation of a function `f` of the product `(X, Y)` is almost everywhere equal
 to the integral of `y ↦ f(X, y)` against the `cond_distrib` kernel. -/
-theorem condexp_prod_ae_eq_integral_cond_distrib' [NormedSpace ℝ F] [CompleteSpace F]
+theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSpace F]
     (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a) :=
@@ -231,11 +268,13 @@ theorem condexp_prod_ae_eq_integral_cond_distrib' [NormedSpace ℝ F] [CompleteS
       set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.ae_measurable.prod_mk hY),
       mk_preimage_prod, preimage_univ, inter_univ]
   · exact ae_strongly_measurable'_integral_cond_distrib hX.ae_measurable hY hf_int.1
-#align probability_theory.condexp_prod_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_prod_ae_eq_integral_cond_distrib'
+#align probability_theory.condexp_prod_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib'
+-/
 
+#print ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib₀ /-
 /-- The conditional expectation of a function `f` of the product `(X, Y)` is almost everywhere equal
 to the integral of `y ↦ f(X, y)` against the `cond_distrib` kernel. -/
-theorem condexp_prod_ae_eq_integral_cond_distrib₀ [NormedSpace ℝ F] [CompleteSpace F]
+theorem condexp_prod_ae_eq_integral_condDistrib₀ [NormedSpace ℝ F] [CompleteSpace F]
     (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a)))
     (hf_int : Integrable (fun a => f (X a, Y a)) μ) :
@@ -243,8 +282,10 @@ theorem condexp_prod_ae_eq_integral_cond_distrib₀ [NormedSpace ℝ F] [Complet
   haveI hf_int' : integrable f (μ.map fun a => (X a, Y a)) := by
     rwa [integrable_map_measure hf (hX.ae_measurable.prod_mk hY)]
   condexp_prod_ae_eq_integral_cond_distrib' hX hY hf_int'
-#align probability_theory.condexp_prod_ae_eq_integral_cond_distrib₀ ProbabilityTheory.condexp_prod_ae_eq_integral_cond_distrib₀
+#align probability_theory.condexp_prod_ae_eq_integral_cond_distrib₀ ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib₀
+-/
 
+#print ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib /-
 /-- The conditional expectation of a function `f` of the product `(X, Y)` is almost everywhere equal
 to the integral of `y ↦ f(X, y)` against the `cond_distrib` kernel. -/
 theorem condexp_prod_ae_eq_integral_condDistrib [NormedSpace ℝ F] [CompleteSpace F]
@@ -255,23 +296,29 @@ theorem condexp_prod_ae_eq_integral_condDistrib [NormedSpace ℝ F] [CompleteSpa
     rwa [integrable_map_measure hf.ae_strongly_measurable (hX.ae_measurable.prod_mk hY)]
   condexp_prod_ae_eq_integral_cond_distrib' hX hY hf_int'
 #align probability_theory.condexp_prod_ae_eq_integral_cond_distrib ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib
+-/
 
+#print ProbabilityTheory.condexp_ae_eq_integral_condDistrib /-
 theorem condexp_ae_eq_integral_condDistrib [NormedSpace ℝ F] [CompleteSpace F] (hX : Measurable X)
     (hY : AEMeasurable Y μ) {f : Ω → F} (hf : StronglyMeasurable f)
     (hf_int : Integrable (fun a => f (Y a)) μ) :
     μ[fun a => f (Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib Y X μ (X a) :=
   condexp_prod_ae_eq_integral_condDistrib hX hY (hf.comp_measurable measurable_snd) hf_int
 #align probability_theory.condexp_ae_eq_integral_cond_distrib ProbabilityTheory.condexp_ae_eq_integral_condDistrib
+-/
 
+#print ProbabilityTheory.condexp_ae_eq_integral_condDistrib' /-
 /-- The conditional expectation of `Y` given `X` is almost everywhere equal to the integral
 `∫ y, y ∂(cond_distrib Y X μ (X a))`. -/
-theorem condexp_ae_eq_integral_cond_distrib' {Ω} [NormedAddCommGroup Ω] [NormedSpace ℝ Ω]
+theorem condexp_ae_eq_integral_condDistrib' {Ω} [NormedAddCommGroup Ω] [NormedSpace ℝ Ω]
     [CompleteSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] {Y : α → Ω}
     (hX : Measurable X) (hY_int : Integrable Y μ) :
     μ[Y|mβ.comap X] =ᵐ[μ] fun a => ∫ y, y ∂condDistrib Y X μ (X a) :=
   condexp_ae_eq_integral_condDistrib hX hY_int.1.AEMeasurable stronglyMeasurable_id hY_int
-#align probability_theory.condexp_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_ae_eq_integral_cond_distrib'
+#align probability_theory.condexp_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_ae_eq_integral_condDistrib'
+-/
 
+#print MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk /-
 theorem MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk {Ω F} {mΩ : MeasurableSpace Ω}
     {X : Ω → β} {μ : Measure Ω} [TopologicalSpace F] (hX : Measurable X) {f : Ω → F}
     (hf : AEStronglyMeasurable f μ) :
@@ -289,7 +336,9 @@ theorem MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk {Ω F} {mΩ : Me
   · exact hX.prod_mk measurable_id
   · exact measurable_snd hs
 #align measure_theory.ae_strongly_measurable.comp_snd_map_prod_mk MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
+-/
 
+#print MeasureTheory.Integrable.comp_snd_map_prod_mk /-
 theorem MeasureTheory.Integrable.comp_snd_map_prod_mk {Ω} {mΩ : MeasurableSpace Ω} {X : Ω → β}
     {μ : Measure Ω} (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
     Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) :=
@@ -299,25 +348,32 @@ theorem MeasureTheory.Integrable.comp_snd_map_prod_mk {Ω} {mΩ : MeasurableSpac
   rw [has_finite_integral, lintegral_map' hf.ennnorm (hX.prod_mk measurable_id).AEMeasurable]
   exact hf_int.2
 #align measure_theory.integrable.comp_snd_map_prod_mk MeasureTheory.Integrable.comp_snd_map_prod_mk
+-/
 
-theorem aEStronglyMeasurable_comp_snd_map_prod_mk_iff {Ω F} {mΩ : MeasurableSpace Ω}
+#print ProbabilityTheory.aestronglyMeasurable_comp_snd_map_prod_mk_iff /-
+theorem aestronglyMeasurable_comp_snd_map_prod_mk_iff {Ω F} {mΩ : MeasurableSpace Ω}
     [TopologicalSpace F] {X : Ω → β} {μ : Measure Ω} (hX : Measurable X) {f : Ω → F} :
     AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔
       AEStronglyMeasurable f μ :=
   ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk hX⟩
-#align probability_theory.ae_strongly_measurable_comp_snd_map_prod_mk_iff ProbabilityTheory.aEStronglyMeasurable_comp_snd_map_prod_mk_iff
+#align probability_theory.ae_strongly_measurable_comp_snd_map_prod_mk_iff ProbabilityTheory.aestronglyMeasurable_comp_snd_map_prod_mk_iff
+-/
 
+#print ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff /-
 theorem integrable_comp_snd_map_prod_mk_iff {Ω} {mΩ : MeasurableSpace Ω} {X : Ω → β} {μ : Measure Ω}
     (hX : Measurable X) {f : Ω → F} :
     Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔ Integrable f μ :=
   ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk hX⟩
 #align probability_theory.integrable_comp_snd_map_prod_mk_iff ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff
+-/
 
+#print ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id /-
 theorem condexp_ae_eq_integral_condDistrib_id [NormedSpace ℝ F] [CompleteSpace F] {X : Ω → β}
     {μ : Measure Ω} [IsFiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
     μ[f|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib id X μ (X a) :=
-  condexp_prod_ae_eq_integral_cond_distrib' hX aemeasurable_id (hf_int.comp_snd_map_prod_mk hX)
+  condexp_prod_ae_eq_integral_condDistrib' hX aemeasurable_id (hf_int.comp_snd_map_prod_mk hX)
 #align probability_theory.condexp_ae_eq_integral_cond_distrib_id ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id
+-/
 
 end ProbabilityTheory
 

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

@@ -72,8 +72,6 @@ instance [MeasurableSpace β] : IsMarkovKernel (condDistrib Y X μ) := by rw [co
 
 variable {mβ : MeasurableSpace β} {s : Set Ω} {t : Set β} {f : β × Ω → F}
 
-include mβ
-
 section Measurability
 
 theorem measurable_condDistrib (hs : MeasurableSet s) :

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

@@ -103,11 +103,11 @@ theorem MeasureTheory.AEStronglyMeasurable.integral_condDistrib (hX : AEMeasurab
   (hf.integral_condDistrib_map hX hY).comp_aemeasurable hX
 #align measure_theory.ae_strongly_measurable.integral_cond_distrib MeasureTheory.AEStronglyMeasurable.integral_condDistrib
 
-theorem aeStronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
+theorem aEStronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
     (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
-    AeStronglyMeasurable' (mβ.comap X) (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
+    AEStronglyMeasurable' (mβ.comap X) (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
   (hf.integral_condDistrib_map hX hY).comp_ae_measurable' hX
-#align probability_theory.ae_strongly_measurable'_integral_cond_distrib ProbabilityTheory.aeStronglyMeasurable'_integral_condDistrib
+#align probability_theory.ae_strongly_measurable'_integral_cond_distrib ProbabilityTheory.aEStronglyMeasurable'_integral_condDistrib
 
 end Measurability
 

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

@@ -120,7 +120,7 @@ theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableS
   · exact Measurable.comp_aemeasurable (kernel.measurable_coe _ hs) hX
   · refine' ne_of_lt _
     calc
-      (∫⁻ a, cond_distrib Y X μ (X a) s ∂μ) ≤ ∫⁻ a, 1 ∂μ := lintegral_mono fun a => prob_le_one
+      ∫⁻ a, cond_distrib Y X μ (X a) s ∂μ ≤ ∫⁻ a, 1 ∂μ := lintegral_mono fun a => prob_le_one
       _ = μ univ := lintegral_one
       _ < ∞ := measure_lt_top _ _
 #align probability_theory.integrable_to_real_cond_distrib ProbabilityTheory.integrable_toReal_condDistrib
@@ -180,7 +180,7 @@ end Integrability
 
 theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hs : MeasurableSet s) (ht : MeasurableSet t) :
-    (∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ) = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by
+    ∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by
   rw [lintegral_comp (kernel.measurable_coe _ hs) hX, cond_distrib, ← measure.restrict_map hX ht, ←
     measure.fst_map_prod_mk₀ hX.ae_measurable hY, set_lintegral_cond_kernel_eq_measure_prod _ ht hs,
     measure.map_apply_of_ae_measurable (hX.ae_measurable.prod_mk hY) (ht.prod hs), mk_preimage_prod]
@@ -188,7 +188,7 @@ theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurabl
 
 theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
     (hs : MeasurableSet s) {t : Set α} (ht : measurable_set[mβ.comap X] t) :
-    (∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ) = μ (t ∩ Y ⁻¹' s) := by obtain ⟨t', ht', rfl⟩ := ht;
+    ∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by obtain ⟨t', ht', rfl⟩ := ht;
   rw [set_lintegral_preimage_cond_distrib hX hY hs ht']
 #align probability_theory.set_lintegral_cond_distrib_of_measurable_set ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet
 
@@ -222,7 +222,7 @@ theorem condexp_prod_ae_eq_integral_cond_distrib' [NormedSpace ℝ F] [CompleteS
   · exact (hf_int.integral_cond_distrib hX.ae_measurable hY).IntegrableOn
   · rintro s ⟨t, ht, rfl⟩ _
     change
-      (∫ a in X ⁻¹' t, ((fun x' => ∫ y, f (x', y) ∂(cond_distrib Y X μ) x') ∘ X) a ∂μ) =
+      ∫ a in X ⁻¹' t, ((fun x' => ∫ y, f (x', y) ∂(cond_distrib Y X μ) x') ∘ X) a ∂μ =
         ∫ a in X ⁻¹' t, f (X a, Y a) ∂μ
     rw [← integral_map hX.ae_measurable]
     swap

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

@@ -123,7 +123,6 @@ theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableS
       (∫⁻ a, cond_distrib Y X μ (X a) s ∂μ) ≤ ∫⁻ a, 1 ∂μ := lintegral_mono fun a => prob_le_one
       _ = μ univ := lintegral_one
       _ < ∞ := measure_lt_top _ _
-      
 #align probability_theory.integrable_to_real_cond_distrib ProbabilityTheory.integrable_toReal_condDistrib
 
 theorem MeasureTheory.Integrable.condDistrib_ae_map (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)

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

@@ -53,7 +53,7 @@ open scoped ENNReal MeasureTheory ProbabilityTheory
 namespace ProbabilityTheory
 
 variable {α β Ω F : Type _} [TopologicalSpace Ω] [MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω]
-  [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : Measure α} [FiniteMeasure μ]
+  [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ]
   {X : α → β} {Y : α → Ω}
 
 /-- **Regular conditional probability distribution**: kernel associated with the conditional
@@ -63,7 +63,7 @@ the conditional expectation `μ⟦Y ⁻¹' s | mβ.comap X⟧ a`. It also satisf
 `μ[(λ a, f (X a, Y a)) | mβ.comap X] =ᵐ[μ] λ a, ∫ y, f (X a, y) ∂(cond_distrib Y X μ (X a))` for
 all integrable functions `f`. -/
 noncomputable irreducible_def condDistrib {mα : MeasurableSpace α} [MeasurableSpace β] (Y : α → Ω)
-  (X : α → β) (μ : Measure α) [FiniteMeasure μ] : kernel β Ω :=
+    (X : α → β) (μ : Measure α) [IsFiniteMeasure μ] : kernel β Ω :=
   (μ.map fun a => (X a, Y a)).condKernel
 #align probability_theory.cond_distrib ProbabilityTheory.condDistrib
 
@@ -317,7 +317,7 @@ theorem integrable_comp_snd_map_prod_mk_iff {Ω} {mΩ : MeasurableSpace Ω} {X :
 #align probability_theory.integrable_comp_snd_map_prod_mk_iff ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff
 
 theorem condexp_ae_eq_integral_condDistrib_id [NormedSpace ℝ F] [CompleteSpace F] {X : Ω → β}
-    {μ : Measure Ω} [FiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
+    {μ : Measure Ω} [IsFiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
     μ[f|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib id X μ (X a) :=
   condexp_prod_ae_eq_integral_cond_distrib' hX aemeasurable_id (hf_int.comp_snd_map_prod_mk hX)
 #align probability_theory.condexp_ae_eq_integral_cond_distrib_id ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id

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

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

@@ -217,7 +217,7 @@ theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEM
 `s` is equal to the conditional expectation of the indicator of `Y ⁻¹' s`. -/
 theorem condDistrib_ae_eq_condexp (hX : Measurable X) (hY : Measurable Y) (hs : MeasurableSet s) :
     (fun a => (condDistrib Y X μ (X a) s).toReal) =ᵐ[μ] μ⟦Y ⁻¹' s|mβ.comap X⟧ := by
-  refine' ae_eq_condexp_of_forall_set_integral_eq hX.comap_le _ _ _ _
+  refine' ae_eq_condexp_of_forall_setIntegral_eq hX.comap_le _ _ _ _
   · exact (integrable_const _).indicator (hY hs)
   · exact fun t _ _ => (integrable_toReal_condDistrib hX.aemeasurable hs).integrableOn
   · intro t ht _
@@ -237,7 +237,7 @@ theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSp
     μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a,y) ∂condDistrib Y X μ (X a) := by
   have hf_int' : Integrable (fun a => f (X a, Y a)) μ :=
     (integrable_map_measure hf_int.1 (hX.aemeasurable.prod_mk hY)).mp hf_int
-  refine' (ae_eq_condexp_of_forall_set_integral_eq hX.comap_le hf_int' (fun s _ _ => _) _ _).symm
+  refine' (ae_eq_condexp_of_forall_setIntegral_eq hX.comap_le hf_int' (fun s _ _ => _) _ _).symm
   · exact (hf_int.integral_condDistrib hX.aemeasurable hY).integrableOn
   · rintro s ⟨t, ht, rfl⟩ _
     change ∫ a in X ⁻¹' t, ((fun x' => ∫ y, f (x', y) ∂(condDistrib Y X μ) x') ∘ X) a ∂μ =
@@ -248,8 +248,8 @@ theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSp
     · rw [← Measure.restrict_map hX ht]
       exact (hf_int.1.integral_condDistrib_map hY).restrict
     rw [← Measure.restrict_map hX ht, ← Measure.fst_map_prod_mk₀ hY, condDistrib,
-      Measure.set_integral_condKernel_univ_right ht hf_int.integrableOn,
-      set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),
+      Measure.setIntegral_condKernel_univ_right ht hf_int.integrableOn,
+      setIntegral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),
       mk_preimage_prod, preimage_univ, inter_univ]
   · exact aestronglyMeasurable'_integral_condDistrib hX.aemeasurable hY hf_int.1
 #align probability_theory.condexp_prod_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib'

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

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

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

@@ -3,7 +3,7 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathlib.Probability.Kernel.Disintegration
+import Mathlib.Probability.Kernel.Disintegration.Unique
 import Mathlib.Probability.Notation
 
 #align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
@@ -74,7 +74,7 @@ variable {mβ : MeasurableSpace β} {s : Set Ω} {t : Set β} {f : β × Ω →
 lemma condDistrib_apply_of_ne_zero [MeasurableSingletonClass β]
     (hY : Measurable Y) (x : β) (hX : μ.map X {x} ≠ 0) (s : Set Ω) :
     condDistrib Y X μ x s = (μ.map X {x})⁻¹ * μ.map (fun a => (X a, Y a)) ({x} ×ˢ s) := by
-  rw [condDistrib, condKernel_apply_of_ne_zero _ s]
+  rw [condDistrib, Measure.condKernel_apply_of_ne_zero _ s]
   · rw [Measure.fst_map_prod_mk hY]
   · rwa [Measure.fst_map_prod_mk hY]
 
@@ -125,7 +125,7 @@ theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measu
     rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
     exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
   rw [heq, condDistrib]
-  refine' eq_condKernel_of_measure_eq_compProd _ _ _
+  refine eq_condKernel_of_measure_eq_compProd _ ?_
   convert hκ
   exact heq.symm
 
@@ -202,7 +202,7 @@ theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurabl
   -- (`rw` does not see that the two forms are defeq)
   conv_lhs => arg 2; change (fun a => ((condDistrib Y X μ) a) s) ∘ X
   rw [lintegral_comp (kernel.measurable_coe _ hs) hX, condDistrib, ← Measure.restrict_map hX ht, ←
-    Measure.fst_map_prod_mk₀ hY, set_lintegral_condKernel_eq_measure_prod _ ht hs,
+    Measure.fst_map_prod_mk₀ hY, Measure.set_lintegral_condKernel_eq_measure_prod ht hs,
     Measure.map_apply_of_aemeasurable (hX.aemeasurable.prod_mk hY) (ht.prod hs), mk_preimage_prod]
 #align probability_theory.set_lintegral_preimage_cond_distrib ProbabilityTheory.set_lintegral_preimage_condDistrib
 
@@ -248,7 +248,7 @@ theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSp
     · rw [← Measure.restrict_map hX ht]
       exact (hf_int.1.integral_condDistrib_map hY).restrict
     rw [← Measure.restrict_map hX ht, ← Measure.fst_map_prod_mk₀ hY, condDistrib,
-      set_integral_condKernel_univ_right ht hf_int.integrableOn,
+      Measure.set_integral_condKernel_univ_right ht hf_int.integrableOn,
       set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),
       mk_preimage_prod, preimage_univ, inter_univ]
   · exact aestronglyMeasurable'_integral_condDistrib hX.aemeasurable hY hf_int.1

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -120,8 +120,8 @@ end Measurability
 theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
     (κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
     ∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by
-  have heq : μ.map X = (μ.map (fun x => (X x, Y x))).fst
-  · ext s hs
+  have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
+    ext s hs
     rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
     exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
   rw [heq, condDistrib]
@@ -300,8 +300,8 @@ theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
     (hf : AEStronglyMeasurable f μ) :
     AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) := by
   refine' ⟨fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable measurable_snd, _⟩
-  suffices h : Measure.QuasiMeasurePreserving Prod.snd (μ.map fun ω => (X ω, ω)) μ
-  · exact Measure.QuasiMeasurePreserving.ae_eq h hf.ae_eq_mk
+  suffices h : Measure.QuasiMeasurePreserving Prod.snd (μ.map fun ω ↦ (X ω, ω)) μ from
+    Measure.QuasiMeasurePreserving.ae_eq h hf.ae_eq_mk
   refine' ⟨measurable_snd, Measure.AbsolutelyContinuous.mk fun s hs hμs => _⟩
   rw [Measure.map_apply _ hs]
   swap; · exact measurable_snd

From the PFR project.

@@ -69,6 +69,15 @@ instance [MeasurableSpace β] : IsMarkovKernel (condDistrib Y X μ) := by
 
 variable {mβ : MeasurableSpace β} {s : Set Ω} {t : Set β} {f : β × Ω → F}
 
+/-- If the singleton `{x}` has non-zero mass for `μ.map X`, then for all `s : Set Ω`,
+`condDistrib Y X μ x s = (μ.map X {x})⁻¹ * μ.map (fun a => (X a, Y a)) ({x} ×ˢ s)` . -/
+lemma condDistrib_apply_of_ne_zero [MeasurableSingletonClass β]
+    (hY : Measurable Y) (x : β) (hX : μ.map X {x} ≠ 0) (s : Set Ω) :
+    condDistrib Y X μ x s = (μ.map X {x})⁻¹ * μ.map (fun a => (X a, Y a)) ({x} ×ˢ s) := by
+  rw [condDistrib, condKernel_apply_of_ne_zero _ s]
+  · rw [Measure.fst_map_prod_mk hY]
+  · rwa [Measure.fst_map_prod_mk hY]
+
 section Measurability
 
 theorem measurable_condDistrib (hs : MeasurableSet s) :

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>

@@ -109,9 +109,7 @@ end Measurability
 /-- `condDistrib` is a.e. uniquely defined as the kernel satisfying the defining property of
 `condKernel`. -/
 theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
-    (κ : kernel β Ω) [IsFiniteKernel κ]
-    (hκ : μ.map (fun x => (X x, Y x)) =
-      (kernel.const Unit (μ.map X) ⊗ₖ kernel.prodMkLeft Unit κ) ()) :
+    (κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
     ∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by
   have heq : μ.map X = (μ.map (fun x => (X x, Y x))).fst
   · ext s hs

@@ -49,7 +49,7 @@ open scoped ENNReal MeasureTheory ProbabilityTheory
 
 namespace ProbabilityTheory
 
-variable {α β Ω F : Type*} [TopologicalSpace Ω] [MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω]
+variable {α β Ω F : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
   [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ]
   {X : α → β} {Y : α → Ω}
 

Co-authored-by: JasonKYi <kexing.ying19@imperial.ac.uk> Co-authored-by: RemyDegenne <Remydegenne@gmail.com>

@@ -106,6 +106,22 @@ theorem aestronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY
 
 end Measurability
 
+/-- `condDistrib` is a.e. uniquely defined as the kernel satisfying the defining property of
+`condKernel`. -/
+theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
+    (κ : kernel β Ω) [IsFiniteKernel κ]
+    (hκ : μ.map (fun x => (X x, Y x)) =
+      (kernel.const Unit (μ.map X) ⊗ₖ kernel.prodMkLeft Unit κ) ()) :
+    ∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by
+  have heq : μ.map X = (μ.map (fun x => (X x, Y x))).fst
+  · ext s hs
+    rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
+    exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
+  rw [heq, condDistrib]
+  refine' eq_condKernel_of_measure_eq_compProd _ _ _
+  convert hκ
+  exact heq.symm
+
 section Integrability
 
 theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :

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

This has nice performance benefits.

@@ -49,7 +49,7 @@ open scoped ENNReal MeasureTheory ProbabilityTheory
 
 namespace ProbabilityTheory
 
-variable {α β Ω F : Type _} [TopologicalSpace Ω] [MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω]
+variable {α β Ω F : Type*} [TopologicalSpace Ω] [MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω]
   [Nonempty Ω] [NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ]
   {X : α → β} {Y : α → Ω}
 

@@ -23,7 +23,7 @@ on `s` can prevent us from finding versions of the conditional expectation that
 measure. The standard Borel space assumption on `Ω` allows us to do so.
 
 The case `Y = X = id` is developed in more detail in `Probability/Kernel/Condexp.lean`: here `X` is
-understood as a map from `Ω` with a sub-σ-algebra to `Ω` with its default σ-algebra and the
+understood as a map from `Ω` with a sub-σ-algebra `m` to `Ω` with its default σ-algebra and the
 conditional distribution defines a kernel associated with the conditional expectation with respect
 to `m`.
 
@@ -77,32 +77,31 @@ theorem measurable_condDistrib (hs : MeasurableSet s) :
 #align probability_theory.measurable_cond_distrib ProbabilityTheory.measurable_condDistrib
 
 theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
-    (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
-    (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
+    (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     (∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
       Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map X) ↔
     Integrable f (μ.map fun a => (X a, Y a)) := by
-  rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prod_mk₀ hX hY]
+  rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prod_mk₀ hY]
 #align measure_theory.ae_strongly_measurable.ae_integrable_cond_distrib_map_iff MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
 
 variable [NormedSpace ℝ F] [CompleteSpace F]
 
-theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map (hX : AEMeasurable X μ)
+theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
     (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) := by
-  rw [← Measure.fst_map_prod_mk₀ hX hY, condDistrib]; exact hf.integral_condKernel
+  rw [← Measure.fst_map_prod_mk₀ hY, condDistrib]; exact hf.integral_condKernel
 #align measure_theory.ae_strongly_measurable.integral_cond_distrib_map MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
 
 theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
-  (hf.integral_condDistrib_map hX hY).comp_aemeasurable hX
+  (hf.integral_condDistrib_map hY).comp_aemeasurable hX
 #align measure_theory.ae_strongly_measurable.integral_cond_distrib MeasureTheory.AEStronglyMeasurable.integral_condDistrib
 
 theorem aestronglyMeasurable'_integral_condDistrib (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
     (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
     AEStronglyMeasurable' (mβ.comap X) (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
-  (hf.integral_condDistrib_map hX hY).comp_ae_measurable' hX
+  (hf.integral_condDistrib_map hY).comp_ae_measurable' hX
 #align probability_theory.ae_strongly_measurable'_integral_cond_distrib ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib
 
 end Measurability
@@ -120,55 +119,55 @@ theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableS
       _ < ∞ := measure_lt_top _ _
 #align probability_theory.integrable_to_real_cond_distrib ProbabilityTheory.integrable_toReal_condDistrib
 
-theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map (hX : AEMeasurable X μ)
+theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     ∀ᵐ b ∂μ.map X, Integrable (fun ω => f (b, ω)) (condDistrib Y X μ b) := by
-  rw [condDistrib, ← Measure.fst_map_prod_mk₀ hX hY]; exact hf_int.condKernel_ae
+  rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.condKernel_ae
 #align measure_theory.integrable.cond_distrib_ae_map MeasureTheory.Integrable.condDistrib_ae_map
 
 theorem _root_.MeasureTheory.Integrable.condDistrib_ae (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     ∀ᵐ a ∂μ, Integrable (fun ω => f (X a, ω)) (condDistrib Y X μ (X a)) :=
-  ae_of_ae_map hX (hf_int.condDistrib_ae_map hX hY)
+  ae_of_ae_map hX (hf_int.condDistrib_ae_map hY)
 #align measure_theory.integrable.cond_distrib_ae MeasureTheory.Integrable.condDistrib_ae
 
-theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map (hX : AEMeasurable X μ)
+theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) := by
-  rw [condDistrib, ← Measure.fst_map_prod_mk₀ hX hY]; exact hf_int.integral_norm_condKernel
+  rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.integral_norm_condKernel
 #align measure_theory.integrable.integral_norm_cond_distrib_map MeasureTheory.Integrable.integral_norm_condDistrib_map
 
 theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ∫ y, ‖f (X a, y)‖ ∂condDistrib Y X μ (X a)) μ :=
-  (hf_int.integral_norm_condDistrib_map hX hY).comp_aemeasurable hX
+  (hf_int.integral_norm_condDistrib_map hY).comp_aemeasurable hX
 #align measure_theory.integrable.integral_norm_cond_distrib MeasureTheory.Integrable.integral_norm_condDistrib
 
 variable [NormedSpace ℝ F] [CompleteSpace F]
 
-theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map (hX : AEMeasurable X μ)
+theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) := by
-  rw [condDistrib, ← Measure.fst_map_prod_mk₀ hX hY]; exact hf_int.norm_integral_condKernel
+  rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.norm_integral_condKernel
 #align measure_theory.integrable.norm_integral_cond_distrib_map MeasureTheory.Integrable.norm_integral_condDistrib_map
 
 theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ‖∫ y, f (X a, y) ∂condDistrib Y X μ (X a)‖) μ :=
-  (hf_int.norm_integral_condDistrib_map hX hY).comp_aemeasurable hX
+  (hf_int.norm_integral_condDistrib_map hY).comp_aemeasurable hX
 #align measure_theory.integrable.norm_integral_cond_distrib MeasureTheory.Integrable.norm_integral_condDistrib
 
-theorem _root_.MeasureTheory.Integrable.integral_condDistrib_map (hX : AEMeasurable X μ)
+theorem _root_.MeasureTheory.Integrable.integral_condDistrib_map
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) :=
-  (integrable_norm_iff (hf_int.1.integral_condDistrib_map hX hY)).mp
-    (hf_int.norm_integral_condDistrib_map hX hY)
+  (integrable_norm_iff (hf_int.1.integral_condDistrib_map hY)).mp
+    (hf_int.norm_integral_condDistrib_map hY)
 #align measure_theory.integrable.integral_cond_distrib_map MeasureTheory.Integrable.integral_condDistrib_map
 
 theorem _root_.MeasureTheory.Integrable.integral_condDistrib (hX : AEMeasurable X μ)
     (hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
     Integrable (fun a => ∫ y, f (X a, y) ∂condDistrib Y X μ (X a)) μ :=
-  (hf_int.integral_condDistrib_map hX hY).comp_aemeasurable hX
+  (hf_int.integral_condDistrib_map hY).comp_aemeasurable hX
 #align measure_theory.integrable.integral_cond_distrib MeasureTheory.Integrable.integral_condDistrib
 
 end Integrability
@@ -180,7 +179,7 @@ theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurabl
   -- (`rw` does not see that the two forms are defeq)
   conv_lhs => arg 2; change (fun a => ((condDistrib Y X μ) a) s) ∘ X
   rw [lintegral_comp (kernel.measurable_coe _ hs) hX, condDistrib, ← Measure.restrict_map hX ht, ←
-    Measure.fst_map_prod_mk₀ hX.aemeasurable hY, set_lintegral_condKernel_eq_measure_prod _ ht hs,
+    Measure.fst_map_prod_mk₀ hY, set_lintegral_condKernel_eq_measure_prod _ ht hs,
     Measure.map_apply_of_aemeasurable (hX.aemeasurable.prod_mk hY) (ht.prod hs), mk_preimage_prod]
 #align probability_theory.set_lintegral_preimage_cond_distrib ProbabilityTheory.set_lintegral_preimage_condDistrib
 
@@ -224,8 +223,8 @@ theorem condexp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSp
     rw [← integral_map hX.aemeasurable (f := fun x' => ∫ y, f (x', y) ∂(condDistrib Y X μ) x')]
     swap
     · rw [← Measure.restrict_map hX ht]
-      exact (hf_int.1.integral_condDistrib_map hX.aemeasurable hY).restrict
-    rw [← Measure.restrict_map hX ht, ← Measure.fst_map_prod_mk₀ hX.aemeasurable hY, condDistrib,
+      exact (hf_int.1.integral_condDistrib_map hY).restrict
+    rw [← Measure.restrict_map hX ht, ← Measure.fst_map_prod_mk₀ hY, condDistrib,
       set_integral_condKernel_univ_right ht hf_int.integrableOn,
       set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),
       mk_preimage_prod, preimage_univ, inter_univ]
@@ -271,8 +270,10 @@ theorem condexp_ae_eq_integral_condDistrib' {Ω} [NormedAddCommGroup Ω] [Normed
   condexp_ae_eq_integral_condDistrib hX hY_int.1.aemeasurable stronglyMeasurable_id hY_int
 #align probability_theory.condexp_ae_eq_integral_cond_distrib' ProbabilityTheory.condexp_ae_eq_integral_condDistrib'
 
-theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk {Ω F} {mΩ: MeasurableSpace Ω}
-    {X : Ω → β} {μ : Measure Ω} [TopologicalSpace F] (hX : Measurable X) {f : Ω → F}
+open MeasureTheory
+
+theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
+    {Ω F} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} [TopologicalSpace F] {f : Ω → F}
     (hf : AEStronglyMeasurable f μ) :
     AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) := by
   refine' ⟨fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable measurable_snd, _⟩
@@ -281,39 +282,47 @@ theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk {Ω F} {m
   refine' ⟨measurable_snd, Measure.AbsolutelyContinuous.mk fun s hs hμs => _⟩
   rw [Measure.map_apply _ hs]
   swap; · exact measurable_snd
-  rw [Measure.map_apply]
-  · rw [← univ_prod, mk_preimage_prod, preimage_univ, univ_inter, preimage_id']
-    exact hμs
-  · exact hX.prod_mk measurable_id
-  · exact measurable_snd hs
+  by_cases hX : AEMeasurable X μ
+  · rw [Measure.map_apply_of_aemeasurable]
+    · rw [← univ_prod, mk_preimage_prod, preimage_univ, univ_inter, preimage_id']
+      exact hμs
+    · exact hX.prod_mk aemeasurable_id
+    · exact measurable_snd hs
+  · rw [Measure.map_of_not_aemeasurable]
+    · simp
+    · contrapose! hX; exact measurable_fst.comp_aemeasurable hX
 #align measure_theory.ae_strongly_measurable.comp_snd_map_prod_mk MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
 
-theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_mk {Ω} {mΩ : MeasurableSpace Ω} {X: Ω → β}
-    {μ : Measure Ω} (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
+theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_mk
+    {Ω} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} {f : Ω → F} (hf_int : Integrable f μ) :
     Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) := by
-  have hf := hf_int.1.comp_snd_map_prod_mk hX
-  refine' ⟨hf, _⟩
-  rw [HasFiniteIntegral, lintegral_map' hf.ennnorm (hX.prod_mk measurable_id).aemeasurable]
-  exact hf_int.2
+  by_cases hX : AEMeasurable X μ
+  · have hf := hf_int.1.comp_snd_map_prod_mk X (mΩ := mΩ) (mβ := mβ)
+    refine' ⟨hf, _⟩
+    rw [HasFiniteIntegral, lintegral_map' hf.ennnorm (hX.prod_mk aemeasurable_id)]
+    exact hf_int.2
+  · rw [Measure.map_of_not_aemeasurable]
+    · simp
+    · contrapose! hX; exact measurable_fst.comp_aemeasurable hX
 #align measure_theory.integrable.comp_snd_map_prod_mk MeasureTheory.Integrable.comp_snd_map_prod_mk
 
 theorem aestronglyMeasurable_comp_snd_map_prod_mk_iff {Ω F} {_ : MeasurableSpace Ω}
     [TopologicalSpace F] {X : Ω → β} {μ : Measure Ω} (hX : Measurable X) {f : Ω → F} :
     AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔
     AEStronglyMeasurable f μ :=
-  ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk hX⟩
+  ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk X⟩
 #align probability_theory.ae_strongly_measurable_comp_snd_map_prod_mk_iff ProbabilityTheory.aestronglyMeasurable_comp_snd_map_prod_mk_iff
 
 theorem integrable_comp_snd_map_prod_mk_iff {Ω} {_ : MeasurableSpace Ω} {X : Ω → β} {μ : Measure Ω}
     (hX : Measurable X) {f : Ω → F} :
     Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔ Integrable f μ :=
-  ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk hX⟩
+  ⟨fun h => h.comp_measurable (hX.prod_mk measurable_id), fun h => h.comp_snd_map_prod_mk X⟩
 #align probability_theory.integrable_comp_snd_map_prod_mk_iff ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff
 
 theorem condexp_ae_eq_integral_condDistrib_id [NormedSpace ℝ F] [CompleteSpace F] {X : Ω → β}
     {μ : Measure Ω} [IsFiniteMeasure μ] (hX : Measurable X) {f : Ω → F} (hf_int : Integrable f μ) :
     μ[f|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f y ∂condDistrib id X μ (X a) :=
-  condexp_prod_ae_eq_integral_condDistrib' hX aemeasurable_id (hf_int.comp_snd_map_prod_mk hX)
+  condexp_prod_ae_eq_integral_condDistrib' hX aemeasurable_id (hf_int.comp_snd_map_prod_mk X)
 #align probability_theory.condexp_ae_eq_integral_cond_distrib_id ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id
 
 end ProbabilityTheory

• 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) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.cond_distrib
-! leanprover-community/mathlib commit 00abe0695d8767201e6d008afa22393978bb324d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Probability.Kernel.Disintegration
 import Mathlib.Probability.Notation
 
+#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
+
 /-!
 # Regular conditional probability distribution