chore: forward port leanprover-community/mathlib#19090 (#5291)

Mathlib/MeasureTheory/Constructions/Prod/Basic.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 
 ! This file was ported from Lean 3 source module measure_theory.constructions.prod.basic
-! leanprover-community/mathlib commit 3b88f4005dc2e28d42f974cc1ce838f0dafb39b8
+! leanprover-community/mathlib commit 00abe0695d8767201e6d008afa22393978bb324d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -834,6 +834,19 @@ instance fst.instIsProbabilityMeasure [IsProbabilityMeasure ρ] : IsProbabilityM
     exact measure_univ
 #align measure_theory.measure.fst.measure_theory.is_probability_measure MeasureTheory.Measure.fst.instIsProbabilityMeasure
 
+theorem fst_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α} (hX : AEMeasurable X μ)
+    (hY : AEMeasurable Y μ) : (μ.map fun a => (X a, Y a)).fst = μ.map X := by
+  ext1 s hs
+  rw [Measure.fst_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_fst hs),
+    Measure.map_apply_of_aemeasurable hX hs, ← prod_univ, mk_preimage_prod, preimage_univ,
+    inter_univ]
+#align measure_theory.measure.fst_map_prod_mk₀ MeasureTheory.Measure.fst_map_prod_mk₀
+
+theorem fst_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α} (hX : Measurable X)
+    (hY : Measurable Y) : (μ.map fun a => (X a, Y a)).fst = μ.map X :=
+  fst_map_prod_mk₀ hX.aemeasurable hY.aemeasurable
+#align measure_theory.measure.fst_map_prod_mk MeasureTheory.Measure.fst_map_prod_mk
+
 /-- Marginal measure on `β` obtained from a measure on `ρ` `α × β`, defined by `ρ.map Prod.snd`. -/
 noncomputable def snd (ρ : Measure (α × β)) : Measure β :=
   ρ.map Prod.snd
@@ -857,6 +870,19 @@ instance snd.instIsProbabilityMeasure [IsProbabilityMeasure ρ] : IsProbabilityM
     exact measure_univ
 #align measure_theory.measure.snd.measure_theory.is_probability_measure MeasureTheory.Measure.snd.instIsProbabilityMeasure
 
+theorem snd_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α} (hX : AEMeasurable X μ)
+    (hY : AEMeasurable Y μ) : (μ.map fun a => (X a, Y a)).snd = μ.map Y := by
+  ext1 s hs
+  rw [Measure.snd_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_snd hs),
+    Measure.map_apply_of_aemeasurable hY hs, ← univ_prod, mk_preimage_prod, preimage_univ,
+    univ_inter]
+#align measure_theory.measure.snd_map_prod_mk₀ MeasureTheory.Measure.snd_map_prod_mk₀
+
+theorem snd_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α} (hX : Measurable X)
+    (hY : Measurable Y) : (μ.map fun a => (X a, Y a)).snd = μ.map Y :=
+  snd_map_prod_mk₀ hX.aemeasurable hY.aemeasurable
+#align measure_theory.measure.snd_map_prod_mk MeasureTheory.Measure.snd_map_prod_mk
+
 end Measure
 
 end MeasureTheory