| import Mathlib |
| import OAI.Analysis.BiholderTransport.Duality.CompactDual |
| import OAI.Analysis.BiholderTransport.Volume.DensityBounds |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Filter MeasureTheory Manifold Bundle |
| open scoped ENNReal NNReal ContDiff Topology BoundedContinuousFunction |
|
|
| namespace WeakMTWTransport |
|
|
| section MeasureContacts |
| variable {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] |
| [MeasurableSpace M] [BorelSpace M] |
|
|
| def contactImage (u v : M → ℝ) (A : Set M) : Set M := |
| {y | ∃ x ∈ A, contactGap u v x y = 0} |
|
|
| omit [CompactSpace M] [Nonempty M] [BorelSpace M] in |
|
|
| lemma dual_minimum_swap {μ ν : Measure M} {u v : M →ᵇ ℝ} |
| (hmin : ∀ a b : M →ᵇ ℝ, (∀ x y, 0 ≤ contactGap a b x y) → |
| dualObjective μ ν (u,v) ≤ dualObjective μ ν (a,b)) : |
| ∀ a b : M →ᵇ ℝ, (∀ x y, 0 ≤ contactGap a b x y) → |
| dualObjective ν μ (v,u) ≤ dualObjective ν μ (a,b) := by |
| intro a b hab |
| have hh := hmin b a (fun x y => by rw [contactGap_symm]; exact hab y x) |
| simpa only [dualObjective,add_comm] using hh |
|
|
| omit [CompactSpace M] [Nonempty M] [BorelSpace M] in |
|
|
| lemma contactImage_ae_eq_preimage {μ : Measure M} {u v : M → ℝ} {S : M → M} |
| (hcontact : ∀ y, contactGap u v (S y) y = 0) |
| (hsingle : ∀ᵐ y ∂μ, ∀ x, contactGap u v x y = 0 → x = S y) |
| (A : Set M) : contactImage u v A =ᵐ[μ] S ⁻¹' A := by |
| filter_upwards [hsingle] with y hy |
| apply propext |
| change (∃ x ∈ A, contactGap u v x y = 0) ↔ S y ∈ A |
| constructor |
| · rintro ⟨x,hx,hxy⟩ |
| simpa only [hy x hxy] using hx |
| · intro hx |
| exact ⟨S y,hx,hcontact y⟩ |
|
|
| omit [CompactSpace M] [Nonempty M] [BorelSpace M] in |
|
|
| lemma contactImage_measure {μ ν : Measure M} {u v : M → ℝ} {S : M → M} |
| (hcontact : ∀ y, contactGap u v (S y) y = 0) |
| (hsingle : ∀ᵐ y ∂ν, ∀ x, contactGap u v x y = 0 → x = S y) |
| (hS : AEMeasurable S ν) (hpush : Measure.map S ν = μ) |
| {A : Set M} (hA : MeasurableSet A) : |
| NullMeasurableSet (contactImage u v A) ν ∧ ν (contactImage u v A) = μ A := by |
| have he := contactImage_ae_eq_preimage hcontact hsingle A |
| refine ⟨?_,?_⟩ |
| · exact (hS.nullMeasurableSet_preimage hA).congr he.symm |
| · rw [measure_congr he,← Measure.map_apply_of_aemeasurable hS hA,hpush] |
|
|
| omit [CompactSpace M] [Nonempty M] [BorelSpace M] in |
|
|
| lemma contactImage_volume_comparison {vol : Measure M} {lam cap : ℝ} |
| {rho0 rho1 u v : M → ℝ} |
| (hrho0 : AdmissibleDensity vol lam cap rho0) |
| (hrho1 : AdmissibleDensity vol lam cap rho1) |
| {A : Set M} (hmass : densityMeasure vol rho1 (contactImage u v A) = |
| densityMeasure vol rho0 A) : |
| ENNReal.ofReal lam * vol A ≤ ENNReal.ofReal cap * vol (contactImage u v A) ∧ |
| ENNReal.ofReal lam * vol (contactImage u v A) ≤ ENNReal.ofReal cap * vol A := by |
| have h0 := densityMeasure_bounds hrho0 |
| have h1 := densityMeasure_bounds hrho1 |
| constructor |
| · calc |
| _ ≤ densityMeasure vol rho0 A := h0.1 A |
| _ = densityMeasure vol rho1 (contactImage u v A) := hmass.symm |
| _ ≤ _ := h1.2 (contactImage u v A) |
| · calc |
| _ ≤ densityMeasure vol rho1 (contactImage u v A) := h1.1 (contactImage u v A) |
| _ = densityMeasure vol rho0 A := hmass |
| _ ≤ _ := h0.2 A |
|
|
| end MeasureContacts |
|
|
| end WeakMTWTransport |
| end |
| end |
| end |
|
|
| end OAI |
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|
