File size: 3,131 Bytes
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 62 63 64 65 66 67 68 | 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
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