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e4d9d2b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.MeasureTheory.Measure.Restrict
namespace OAI
/-! # Lifting an ambient coupling to the prescribed measurable supports -/
open Set Filter MeasureTheory
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MeasurableSpace X] {S T : Set X}
theorem coupling_lift_subtypes (hS : MeasurableSet S) (hT : MeasurableSet T)
(μ : Measure S) (ν : Measure T) (π : Measure (X × X))
(hl : π.map Prod.fst=μ.map Subtype.val) (hr : π.map Prod.snd=ν.map Subtype.val) :
∃ Q : Measure (S × T), Q.map (Prod.map Subtype.val Subtype.val)=π ∧
Q.map Prod.fst=μ ∧ Q.map Prod.snd=ν := by
let inc : S × T → X × X := Prod.map Subtype.val Subtype.val
have he : MeasurableEmbedding inc :=
(MeasurableEmbedding.subtype_coe hS).prodMap (MeasurableEmbedding.subtype_coe hT)
have ha : ∀ᵐ z ∂π, z.1 ∈ S := by
apply ae_of_ae_map measurable_fst.aemeasurable
rw [hl]
exact (MeasurableEmbedding.subtype_coe hS).ae_map_iff.mpr
(Eventually.of_forall fun x => x.property)
have hb : ∀ᵐ z ∂π, z.2 ∈ T := by
apply ae_of_ae_map measurable_snd.aemeasurable
rw [hr]
exact (MeasurableEmbedding.subtype_coe hT).ae_map_iff.mpr
(Eventually.of_forall fun x => x.property)
have hm : (π.comap inc).map inc=π := by
rw [he.map_comap]
apply Measure.restrict_eq_self_of_ae_mem
filter_upwards [ha,hb] with z hz ht
exact ⟨(⟨z.1,hz⟩,⟨z.2,ht⟩),rfl⟩
refine ⟨π.comap inc,hm,?_,?_⟩
· apply (MeasurableEmbedding.subtype_coe hS).map_injective
calc
((π.comap inc).map Prod.fst).map Subtype.val = ((π.comap inc).map inc).map Prod.fst := by
rw [Measure.map_map measurable_subtype_coe measurable_fst,
Measure.map_map measurable_fst he.measurable]
rfl
_ = π.map Prod.fst := congrArg (fun m : Measure (X × X) => m.map Prod.fst) hm
_ = μ.map Subtype.val := hl
· apply (MeasurableEmbedding.subtype_coe hT).map_injective
calc
((π.comap inc).map Prod.snd).map Subtype.val = ((π.comap inc).map inc).map Prod.snd := by
rw [Measure.map_map measurable_subtype_coe measurable_snd,
Measure.map_map measurable_snd he.measurable]
rfl
_ = π.map Prod.snd := congrArg (fun m : Measure (X × X) => m.map Prod.snd) hm
_ = ν.map Subtype.val := hr
end RegularPointCoordinates
end
end OAI
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