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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 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
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