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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 60 61 | import OAI.Geometry.RCD.Transport.PairedInitialUniqueness
import OAI.Geometry.RCD.Heat.ConditionalPairInverse
namespace OAI
/-! # Measurable inversion of initial evaluation for the actual compressed plan -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped ENNReal NNReal Topology
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] [SeparableSpace X] {K : ℝ} {n : ℕ}
[IsLocallyFiniteMeasure (hausdorff n X)] (hX : NoncollapsedRCD K n X)
include hX
attribute [local instance] pathBorel pathBorelSpace
theorem compressed_initial_inverse (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R)
(π : Measure C(Time,X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ)
(hp : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1))
(π.map (fun γ => (γ 0,γ 1))))
{C : ℝ≥0} (hC : 0 < C)
(hc : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X)
(hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R)
(hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R)
:
∃ g : X → C(Time,X), Measurable g ∧ ∀ᵐ γ ∂π, g (γ 0)=γ := by
let : CompleteSpace X := hX.complete
let : Nonempty C(Time,X) := nonempty_of_isProbabilityMeasure π
let σ := conditionalSelfPair π (fun γ => γ 0) (measurable_path_eval 0)
obtain ⟨hσ₀,hσ₁⟩ := conditionalSelfPair_marginals π (fun γ => γ 0) (measurable_path_eval 0)
let : IsProbabilityMeasure σ := ⟨by
have h := congrArg (fun m : Measure C(Time,X) => m univ) hσ₀
simpa only [Measure.map_apply measurable_fst MeasurableSet.univ,preimage_univ,measure_univ] using h⟩
have hmeet : ∀ᵐ z ∂σ, z.1 0=z.2 0 :=
conditionalSelfPair_common π (fun γ => γ 0) (measurable_path_eval 0)
have hdiag := hX.paired_common_initial_unique hn p hR π hg hp hC hc hs₀ hs₁
σ hσ₀.le hσ₁.le hmeet
exact measurable_inverse_of_conditional_diagonal π (fun γ => γ 0) (measurable_path_eval 0) hdiag
theorem compressed_initial_injective (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R)
(π : Measure C(Time,X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ)
(hp : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1))
(π.map (fun γ => (γ 0,γ 1))))
{C : ℝ≥0} (hC : 0 < C)
(hc : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X)
(hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R)
(hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R)
:
∃ S : Set C(Time,X), MeasurableSet S ∧ (∀ᵐ γ ∂π, γ ∈ S) ∧ InjOn (fun γ => γ 0) S := by
obtain ⟨g,hg,hgπ⟩ := hX.compressed_initial_inverse hn p hR π hg hp hC hc hs₀ hs₁
refine ⟨{γ | g (γ 0)=γ},measurableSet_eq_fun (hg.comp (measurable_path_eval 0)) measurable_id,hgπ,?_⟩
intro γ hγ ζ hζ he
exact hγ.symm.trans ((congrArg g he).trans hζ)
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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