| import OAI.Geometry.RCD.Transport.LocalizedPlanEnergy |
| import Mathlib.MeasureTheory.Integral.Average |
|
|
| namespace OAI |
|
|
| /-! # Integrating a local defect along a compressed geodesic plan |
|
|
| This is the first-moment step for the functional almost-splitting argument. |
| It retains the full local integrable defect, and permits the selected segment |
| to avoid any additional null set of paths. |
| -/ |
|
|
| open Set Filter Metric MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| theorem compressed_segment_integral |
| (μ : Measure X) (π : Measure C(Time,X)) [IsProbabilityMeasure π] |
| {C : ℝ≥0∞} (hC : C ≠ ∞) |
| (hc : ∀ t : Time, π.map (fun γ => γ t) ≤ C • μ) |
| {Ω : Set X} |
| (hstay : ∀ t : Time, ∀ᵐ x ∂π.map (fun γ => γ t), x ∈ Ω) |
| {F : X → ℝ} (hF : IntegrableOn F Ω μ) |
| (hFpos : ∀ᵐ x ∂μ.restrict Ω, 0 ≤ F x) : |
| Integrable (fun γ : C(Time,X) => |
| ∫ t in Icc (0:ℝ) 1, F (geodesicExtension γ t)) π ∧ |
| (∀ᵐ γ ∂π, IntegrableOn (fun t => F (geodesicExtension γ t)) (Icc (0:ℝ) 1)) ∧ |
| (∫ γ : C(Time,X), (∫ t in Icc (0:ℝ) 1, |
| F (geodesicExtension γ t)) ∂π) ≤ C.toReal * ∫ x in Ω, F x ∂μ := by |
| have hlocal (t : Time) : π.map (fun γ => γ t) ≤ C • μ.restrict Ω := |
| (compression_iff_localized μ _ (hstay t) C).mp (hc t) |
| have hcomp := testPlanMeasure_compression (μ.restrict Ω) π hlocal |
| have hmap := Integrable.of_measure_le_smul hC hcomp hF |
| have hprod := hmap.comp_measurable testPlanEval_measurable |
| have hbound : (∫ z, F (testPlanEval z) ∂testPlanMeasure π) ≤ |
| C.toReal * ∫ x in Ω, F x ∂μ := by |
| rw [← integral_map testPlanEval_measurable.aemeasurable hmap.aestronglyMeasurable] |
| simpa only [integral_smul_measure,smul_eq_mul] using |
| integral_mono_measure hcomp (Measure.ae_smul_measure hFpos C) (hF.smul_measure hC) |
| refine ⟨hprod.integral_prod_right,hprod.prod_left_ae,?_⟩ |
| exact (integral_prod_symm _ hprod).symm.le.trans hbound |
|
|
| theorem exists_compressed_segment |
| (μ : Measure X) (π : Measure C(Time,X)) [IsProbabilityMeasure π] |
| {C : ℝ≥0∞} (hC : C ≠ ∞) |
| (hc : ∀ t : Time, π.map (fun γ => γ t) ≤ C • μ) |
| {Ω : Set X} |
| (hstay : ∀ t : Time, ∀ᵐ x ∂π.map (fun γ => γ t), x ∈ Ω) |
| {F : X → ℝ} (hF : IntegrableOn F Ω μ) |
| (hFpos : ∀ᵐ x ∂μ.restrict Ω, 0 ≤ F x) |
| {P : C(Time,X) → Prop} (hP : ∀ᵐ γ ∂π, P γ) : |
| ∃ γ : C(Time,X), P γ ∧ |
| IntegrableOn (fun t => F (geodesicExtension γ t)) (Icc (0:ℝ) 1) ∧ |
| (∫ t in Icc (0:ℝ) 1, F (geodesicExtension γ t)) ≤ |
| C.toReal * ∫ x in Ω, F x ∂μ := by |
| obtain ⟨hi,hinner,hbound⟩ := compressed_segment_integral μ π hC hc hstay hF hFpos |
| obtain ⟨γ,hγ,havg⟩ := exists_notMem_null_le_integral hi (ae_iff.mp (hP.and hinner)) |
| exact ⟨γ,(of_not_not hγ).1,(of_not_not hγ).2,havg.trans hbound⟩ |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
