| import Mathlib.MeasureTheory.Measure.Prod |
| import Mathlib.MeasureTheory.Measure.Restrict |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| 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 |
|
|
