| import OAI.Geometry.RCD.Transport.PairedPathUniqueness | |
| import OAI.Geometry.RCD.Transport.PairedSuffixUniqueness | |
| namespace OAI | |
| /-! # Explicit optimal half mixtures with their compression and domination -/ | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [SeparableSpace X] | |
| attribute [local instance] pathBorel pathBorelSpace | |
| omit [SeparableSpace X] in | |
| theorem half_mixture_plan_data {ν₀ ν₁ : Measure X} | |
| (π ρ : Measure C(Time,X)) [IsProbabilityMeasure π] [IsProbabilityMeasure ρ] | |
| (hgπ : ∀ᵐ γ ∂π, IsGeodesic γ) (hgρ : ∀ᵐ γ ∂ρ, IsGeodesic γ) | |
| (hpπ : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0,γ 1)))) | |
| (hpρ : IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0,γ 1)))) : | |
| let τ := (2:ℝ≥0∞)⁻¹ • (π+ρ) | |
| IsProbabilityMeasure τ ∧ (∀ᵐ γ ∂τ, IsGeodesic γ) ∧ | |
| IsOptimalCoupling ν₀ ν₁ (τ.map (fun γ => (γ 0,γ 1))) ∧ | |
| (2:ℝ≥0∞)⁻¹ • π ≤ τ ∧ (2:ℝ≥0∞)⁻¹ • ρ ≤ τ := by | |
| let c : ℝ≥0∞ := (2:ℝ≥0∞)⁻¹ | |
| let τ := c • (π+ρ) | |
| have hcc : c+c=1 := by | |
| dsimp only [c] | |
| rw [←two_mul,ENNReal.mul_inv_cancel (by norm_num) (by norm_num)] | |
| have hprob : IsProbabilityMeasure τ := ⟨by | |
| change c*(π univ+ρ univ)=1 | |
| simp only [measure_univ,mul_add,mul_one,hcc]⟩ | |
| have hm := (measurable_path_eval (X:=X) 0).prodMk (measurable_path_eval 1) | |
| have he (ν : Measure X) : c • (ν+ν)=ν := half_smul_self ν | |
| have hopt : IsOptimalCoupling ν₀ ν₁ (τ.map (fun γ => (γ 0,γ 1))) := by | |
| rw [show τ=c • (π+ρ) from rfl,Measure.map_smul _ hm.aemeasurable,Measure.map_add _ _ hm] | |
| refine ⟨?_,?_,?_⟩ | |
| · rw [Measure.map_smul _ measurable_fst.aemeasurable,Measure.map_add _ _ measurable_fst, | |
| hpπ.1,hpρ.1,he] | |
| · rw [Measure.map_smul _ measurable_snd.aemeasurable,Measure.map_add _ _ measurable_snd, | |
| hpπ.2.1,hpρ.2.1,he] | |
| · intro η hη₀ hη₁ | |
| rw [transportCost_smul,transportCost_add,mul_add] | |
| calc | |
| _ ≤ c*transportCost η+c*transportCost η := | |
| add_le_add (mul_le_mul' le_rfl (hpπ.2.2 η hη₀ hη₁)) | |
| (mul_le_mul' le_rfl (hpρ.2.2 η hη₀ hη₁)) | |
| _ = _ := by rw [←add_mul,hcc,one_mul] | |
| refine ⟨hprob,?_,hopt,?_,?_⟩ | |
| · exact ((ae_add_measure_iff.mpr ⟨hgπ,hgρ⟩).filter_mono Measure.smul_absolutelyContinuous.ae_le) | |
| · exact smul_le_smul_left c (Measure.le_add_right le_rfl) | |
| · exact smul_le_smul_left c (Measure.le_add_left le_rfl) | |
| omit [SeparableSpace X] in | |
| theorem half_mixture_compression (μ : Measure X) (π ρ : Measure C(Time,X)) | |
| {C : ℝ≥0∞} (hπ : ∀ t : Time, π.map (fun γ => γ t) ≤ C • μ) | |
| (hρ : ∀ t : Time, ρ.map (fun γ => γ t) ≤ C • μ) (t : Time) : | |
| ((2:ℝ≥0∞)⁻¹ • (π+ρ)).map (fun γ => γ t) ≤ C • μ := by | |
| rw [half_mixture_map_eval] | |
| exact (_root_.smul_le_smul_left ((2:ℝ≥0∞)⁻¹) (add_le_add (hπ t) (hρ t))).trans_eq (half_smul_self _) | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |