| import OAI.Geometry.RCD.Transport.CompactCouplingLift |
| import OAI.Analysis.RCDHeat.Foundations |
| import Mathlib.MeasureTheory.Integral.Bochner.Set |
|
|
| namespace OAI |
|
|
| /-! # Compact subtype cost minimizers are ambient optimal couplings -/ |
|
|
| open Set Filter MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [SecondCountableTopology X] |
| {S T : Set X} [CompactSpace S] [CompactSpace T] |
|
|
| theorem compact_coupling_transportCost (Q : Measure (S × T)) [IsFiniteMeasure Q] : |
| transportCost (Q.map (Prod.map Subtype.val Subtype.val)) = |
| ENNReal.ofReal (∫ z, dist (z.1 : X) (z.2 : X)^2 ∂Q) := by |
| have hm : Measurable (fun z : X × X => ENNReal.ofReal (dist z.1 z.2)^2) := by fun_prop |
| rw [transportCost,lintegral_map hm (measurable_subtype_coe.prodMap measurable_subtype_coe)] |
| have hi : Integrable (fun z : S × T => dist (z.1 : X) (z.2 : X)^2) Q := |
| (by fun_prop : Continuous (fun z : S × T => dist (z.1 : X) (z.2 : X)^2)).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| rw [ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall fun _ => sq_nonneg _)] |
| apply lintegral_congr |
| intro z |
| exact (ENNReal.ofReal_pow dist_nonneg 2).symm |
|
|
| theorem compact_coupling_ambient_optimal (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (μ : Measure S) (ν : Measure T) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (Q : Measure (S × T)) [IsProbabilityMeasure Q] |
| (hl : Q.map Prod.fst=μ) (hr : Q.map Prod.snd=ν) |
| (hmin : ∀ P : Measure (S × T), P.map Prod.fst=μ → P.map Prod.snd=ν → |
| (∫ z, dist (z.1 : X) (z.2 : X)^2 ∂Q) ≤ ∫ z, dist (z.1 : X) (z.2 : X)^2 ∂P) : |
| IsOptimalCoupling (μ.map Subtype.val) (ν.map Subtype.val) |
| (Q.map (Prod.map Subtype.val Subtype.val)) := by |
| have hm : Measurable (Prod.map (Subtype.val : S → X) (Subtype.val : T → X)) := |
| measurable_subtype_coe.prodMap measurable_subtype_coe |
| refine ⟨?_,?_,?_⟩ |
| · rw [Measure.map_map measurable_fst hm,← hl, |
| Measure.map_map measurable_subtype_coe measurable_fst] |
| rfl |
| · rw [Measure.map_map measurable_snd hm,← hr, |
| Measure.map_map measurable_subtype_coe measurable_snd] |
| rfl |
| · intro π hπl hπr |
| obtain ⟨P,hP,hPl,hPr⟩ := coupling_lift_subtypes hS hT μ ν π hπl hπr |
| let : IsProbabilityMeasure (P.map Prod.fst) := by rw [hPl]; infer_instance |
| let : IsProbabilityMeasure P := Measure.isProbabilityMeasure_of_map measurable_fst.aemeasurable |
| rw [← hP,compact_coupling_transportCost Q,compact_coupling_transportCost P] |
| exact ENNReal.ofReal_le_ofReal (hmin P hPl hPr) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
