| import OAI.Geometry.RCD.Transport.BoundedOptimalRecovery |
|
|
| namespace OAI |
|
|
| /-! # Transport nodes of bounded recovered endpoints -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [SeparableSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| def boundedEndpointNode (n : ℕ) (ρ : ProbabilityMeasure C(Time,X)) (p : X) |
| {R : ℝ} (hR : 0≤R) |
| (hs : ∀ᵐ γ ∂(ρ : Measure C(Time,X)), ∀ t : Time, dist (γ t) p≤R) |
| {C : ℝ≥0∞} (hC : C≠∞) |
| (h0 : (ρ : Measure C(Time,X)).map (fun γ => γ 0)≤C • hausdorff n X) |
| (h1 : (ρ : Measure C(Time,X)).map (fun γ => γ 1)≤C • hausdorff n X) : |
| RenyiTransportNode n X := by |
| have he (t : Time) : ∀ᵐ x ∂(ρ : Measure C(Time,X)).map (fun γ => γ t), dist x p≤R := by |
| apply (ae_map_iff (measurable_path_eval t).aemeasurable measurableSet_closedBall).mpr |
| exact hs.mono fun _ hγ => hγ t |
| exact ⟨(ρ : Measure C(Time,X)).map (fun γ => γ 0), |
| (ρ : Measure C(Time,X)).map (fun γ => γ 1),inferInstance,inferInstance, |
| p,R,hR,he 0,he 1,C,hC,h0,h1,2*R,mul_nonneg (by norm_num) hR, |
| optimalPlanDistanceBound_of_bounded _ _ p (he 0) (he 1)⟩ |
|
|
| omit [SeparableSpace X] in |
| theorem bounded_recovery_endpoint_bound |
| (n : ℕ) (ρ τ : Measure C(Time,X)) |
| (hle : τ≤(2 : ℝ≥0∞) • ρ) {C : ℝ≥0∞} (t : Time) |
| (hc : ρ.map (fun γ => γ t)≤C • hausdorff n X) : |
| τ.map (fun γ => γ t)≤((2 : ℝ≥0∞)*C) • hausdorff n X := by |
| have hh := Measure.map_mono hle (measurable_path_eval t) |
| rw [Measure.map_smul _ (measurable_path_eval t).aemeasurable] at hh |
| intro B |
| exact (hh B).trans (by |
| change (2 : ℝ≥0∞)*(ρ.map (fun γ => γ t)) B≤((2 : ℝ≥0∞)*C)*(hausdorff n X) B |
| have hcb : (ρ.map (fun γ => γ t)) B≤C*(hausdorff n X) B := hc B |
| exact (mul_le_mul' (le_refl (2 : ℝ≥0∞)) hcb).trans_eq (mul_assoc 2 C _).symm) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
