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namespace OAI
/-! # An actual compressed midpoint with unequal endpoint bounds
Minimizing excess mass and replacing its saturated region turns the CD
support estimate into a density bound. Separate endpoint bounds allow
one marginal to concentrate while retaining the contribution of the other.
-/
open Set Filter Metric MeasureTheory TopologicalSpace
open scoped ENNReal
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
attribute [local instance] pathBorel pathBorelSpace
theorem FullCD.improve_localized_saturation_asymmetric
(μ : Measure X) [SigmaFinite μ]
(W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)]
(K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X)
[IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁]
{C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0)
(h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ)
(hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁)
(s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁)
(hn : 2 ≤ n) (D : ℝ)
(hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D)
(hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q →
(∀ᵐ γ ∂q, IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) →
∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W)
(hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁)
(π : Measure C(Time, X)) [IsProbabilityMeasure π]
(hg : ∀ᵐ γ ∂π, IsGeodesic γ)
(hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1))))
(hpos : transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≠ 0)
(hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' →
(∀ᵐ γ ∂π', IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) →
transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤
transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) :
∃ π' : Measure C(Time, X), IsProbabilityMeasure π' ∧
(∀ᵐ γ ∂π', IsGeodesic γ) ∧
IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) ∧
transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C =
transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ∧
(μ.restrict W) {x | C < (π.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} +
transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C / 2 * C⁻¹ <
(μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} := by
let ν := π.map (fun γ => γ midpointTime)
have hvprob : IsProbabilityMeasure ν :=
inferInstance
let e := transportExcessMass (μ.restrict W) ν C
let v := (μ.restrict W) {x | C < ν.rnDeriv (μ.restrict W) x}
have hetop : e ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top ν univ)
(transportExcessMass_le (μ.restrict W) ν C)
obtain ⟨s, hs, hμs, hνs⟩ := (Measure.mutuallySingular_singularPart ν (μ.restrict W)).symm
have hthreshold : e / 2 + C * v < ν (s ∪ {x | C < ν.rnDeriv (μ.restrict W) x}) := by
rw [positive_region_mass (μ.restrict W) ν hC hs hμs hνs]
exact ENNReal.add_lt_add_right
(ENNReal.mul_ne_top hC (measure_ne_top (μ.restrict W) _)) (ENNReal.half_lt_self hpos hetop)
obtain ⟨δ, hδ0, hδtop, hδmass⟩ :=
exists_buffer_positive_region (μ.restrict W) ν hC s hthreshold
let B := s ∪ {x | C + δ < ν.rnDeriv (μ.restrict W) x}
have hB : MeasurableSet B := hs.union
(measurableSet_lt measurable_const (ν.measurable_rnDeriv (μ.restrict W)))
have hB0 : ν B ≠ 0 := (lt_of_le_of_lt zero_le hδmass).ne'
let a := δ / (C + δ)
have ha : a ≤ 1 := ENNReal.div_le_of_le_mul
(by simp)
have ha0 : 0 < a := ENNReal.div_pos hδ0.ne' (ENNReal.add_ne_top.mpr ⟨hC, hδtop⟩)
have hrem : a • ν.restrict B ≤ ν - C • μ.restrict W :=
buffer_fraction_le_excess (μ.restrict W) ν hC hδ0 hδtop hs hμs
let : IsFiniteMeasure (a • ν.restrict B) :=
isFiniteMeasure_of_le ν (hrem.trans Measure.sub_le)
obtain ⟨π', η, hπ', hη, hg', hp', heq, hmass, hvol⟩ :=
hcd.replace_midpoint_region_asymmetric μ K n ν₀ ν₁ hC₀ hC₁ h0 h1 hm0 hm1 s₀ s₁ hs0 hs1
hn D hD π hg hp B hB hB0 ha0 ha
let := hπ'
let := hη
have hηle : η ≤ π'.map (fun γ => γ midpointTime) := by
rw [heq]
exact Measure.le_add_left le_rfl
have hηsupp : ∀ᵐ x ∂η, x ∈ W :=
(hsupport π' hπ' hg' hp').filter_mono hηle.absolutelyContinuous.ae_le
have hvolLoc := hvol.trans_eq (ac_support_volume_restrict μ η hW hηsupp)
have hmin' := hmin π' hπ' hg' hp'
have hmass' : η univ = (a • ν.restrict B) univ := by
simpa only [Measure.smul_apply, smul_eq_mul, Measure.restrict_apply_univ] using hmass
have hequal : transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C = e := by
apply le_antisymm _ hmin'
rw [heq]
exact (transportExcessMass_add_le (μ.restrict W) (ν - a • ν.restrict B) η hC).trans_eq (by
rw [hmass', transportExcessMass_remove (μ.restrict W) ν (a • ν.restrict B) hC hrem])
have hsaturated : ∀ᵐ x ∂μ.restrict W, η.rnDeriv (μ.restrict W) x ≠ 0 →
C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x := by
rw [heq]
apply ac_support_saturated_of_minimizing_replacement (μ.restrict W) ν (a • ν.restrict B) η
hC hrem hmass'
simpa only [heq] using hmin'
refine ⟨π', hπ', hg', hp', hequal, ?_⟩
have hgain : v + e / 2 * C⁻¹ < ν B * C⁻¹ := by
have hh := ENNReal.mul_lt_mul_left (ENNReal.inv_ne_zero.mpr hC)
(ENNReal.inv_ne_top.mpr hCpos) hδmass
convert hh using 1
rw [add_mul, mul_assoc, mul_comm v, ← mul_assoc, ENNReal.mul_inv_cancel hCpos hC]
simp only [one_mul, add_comm]
exact hgain.trans_le (calc
ν B * C⁻¹ ≤ ν B * (midpointSupportFactor K n D C₀ C₁) :=
mul_le_mul_right hfill _
_ = midpointSupportFactor K n D C₀ C₁ * ν B := by ac_rfl
_ ≤ (μ.restrict W) {x | η.rnDeriv (μ.restrict W) x ≠ 0} := hvolLoc
_ ≤ (μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} :=
measure_mono_ae hsaturated)
theorem FullCD.localized_excess_minimizer_zero_asymmetric
(μ : Measure X) [SigmaFinite μ]
(W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)]
(K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X)
[IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁]
{C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0)
(h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ)
(hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁)
(s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁)
(hn : 2 ≤ n) (D : ℝ)
(hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D)
(hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q →
(∀ᵐ γ ∂q, IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) →
∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W)
(hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁)
(π : Measure C(Time, X)) [IsProbabilityMeasure π]
(hg : ∀ᵐ γ ∂π, IsGeodesic γ)
(hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1))))
(hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' →
(∀ᵐ γ ∂π', IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) →
transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤
transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) :
transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C = 0 := by
by_contra hpos
let e := transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C
let A := {ρ : Measure C(Time, X) // IsProbabilityMeasure ρ ∧
(∀ᵐ γ ∂ρ, IsGeodesic γ) ∧
IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0, γ 1))) ∧
transportExcessMass (μ.restrict W) (ρ.map (fun γ => γ midpointTime)) C = e}
have : Nonempty A := ⟨⟨π, inferInstance, hg, hp, rfl⟩⟩
let v (ρ : A) := (μ.restrict W) {x | C < (ρ.val.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x}
let S := ⨆ ρ : A, v ρ
have hS : S ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top (μ.restrict W) univ)
(iSup_le fun _ => measure_mono (subset_univ _))
have hstep : S + e / 2 * C⁻¹ ≤ S := by
rw [ENNReal.iSup_add]
apply iSup_le
intro ρ
let : IsProbabilityMeasure ρ.val := ρ.property.1
have hposρ : transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≠ 0 := by
rw [ρ.property.2.2.2]
exact hpos
have hminρ : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' →
(∀ᵐ γ ∂π', IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) →
transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≤
transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C := by
intro π' hπ' hg' hp'
rw [ρ.property.2.2.2]
exact hmin π' hπ' hg' hp'
obtain ⟨π', hπ', hg', hp', hequal, hgain⟩ :=
hcd.improve_localized_saturation_asymmetric μ W hW K n ν₀ ν₁
hC₀ hC₁ hC hCpos h0 h1 hm0 hm1 s₀ s₁ hs0 hs1 hn D hD hsupport hfill
ρ.val ρ.property.2.1 ρ.property.2.2.1 hposρ hminρ
let ρ' : A := ⟨π', hπ', hg', hp', hequal.trans ρ.property.2.2.2⟩
rw [ρ.property.2.2.2] at hgain
exact hgain.le.trans (le_iSup v ρ')
have hδ : 0 < e / 2 * C⁻¹ := ENNReal.mul_pos
(ENNReal.div_pos hpos (by simp)).ne' (ENNReal.inv_ne_zero.mpr hC)
have hstrict := ENNReal.add_lt_add_left hS hδ
rw [add_zero] at hstrict
exact (not_lt_of_ge hstep) hstrict
section Attainment
variable [ProperSpace X] [SeparableSpace X]
theorem FullCD.exists_compressed_midpoint_asymmetric
(μ : Measure X) [SigmaFinite μ] (K : ℝ) (n : ℕ) (hcd : FullCD μ K n)
(ν₀ ν₁ : Measure X) [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁]
{C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0)
(h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ)
(hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁)
(p : X) {R : ℝ} (hR : 0 ≤ R)
[IsFiniteMeasure (μ.restrict (closedBall p (3 * R)))]
[Measure.OuterRegular (μ.restrict (closedBall p (3 * R)))]
(hs0 : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs1 : ∀ᵐ x ∂ν₁, dist x p ≤ R)
(hn : 2 ≤ n)
(hfill : C⁻¹ ≤ midpointSupportFactor K n (2 * R) C₀ C₁) :
∃ π : Measure C(Time, X), IsProbabilityMeasure π ∧
(∀ᵐ γ ∂π, IsGeodesic γ) ∧
IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1))) ∧
π.map (fun γ => γ midpointTime) ≤ C • μ := by
let W := closedBall p (3 * R)
have hW : MeasurableSet W := isClosed_closedBall.measurableSet
have hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q →
(∀ᵐ γ ∂q, IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) →
∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W := by
intro q hq hgq hpq
apply (ae_map_iff (measurable_path_eval midpointTime).aemeasurable hW).mpr
exact (optimal_geodesic_plan_bounded ν₀ ν₁ q hgq hpq p hR hs0 hs1).mono
(fun γ hγ => hγ.2 midpointTime)
let v0 : ProbabilityMeasure X := ⟨ν₀, inferInstance⟩
let v1 : ProbabilityMeasure X := ⟨ν₁, inferInstance⟩
obtain ⟨q, hq, hgq, hpq, _⟩ := hcd ν₀ ν₁ inferInstance inferInstance
(Measure.absolutelyContinuous_of_le_smul h0)
(Measure.absolutelyContinuous_of_le_smul h1) hm0 hm1
have hne : (OptimalGeodesicPlans v0 v1).Nonempty := ⟨⟨q, hq⟩, hgq, hpq⟩
obtain ⟨π, hπ, hmπ⟩ := exists_optimal_geodesic_plan_minimizing_excess
(μ.restrict W) v0 v1 p hR hs0 hs1 hne midpointTime hC
have hmin : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q →
(∀ᵐ γ ∂q, IsGeodesic γ) →
IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) →
transportExcessMass (μ.restrict W) ((π : Measure C(Time, X)).map (fun γ => γ midpointTime)) C ≤
transportExcessMass (μ.restrict W) (q.map (fun γ => γ midpointTime)) C := by
intro q hq hgq hpq
exact hmπ ⟨q, hq⟩ ⟨hgq, hpq⟩
have hD : ∀ x ∈ closedBall p R, ∀ y ∈ closedBall p R, dist x y ≤ 2 * R := by
intro x hx y hy
have hd := dist_triangle x p y
rw [dist_comm p y] at hd
change dist x p ≤ R at hx
change dist y p ≤ R at hy
linarith
have hz := hcd.localized_excess_minimizer_zero_asymmetric μ W hW K n ν₀ ν₁ hC₀ hC₁ hC hCpos
h0 h1 hm0 hm1 (closedBall p R) (closedBall p R) hs0 hs1 hn (2 * R) hD
hsupport hfill (π : Measure C(Time, X)) hπ.1 hπ.2 hmin
refine ⟨π, inferInstance, hπ.1, hπ.2, ?_⟩
have hs := hsupport (π : Measure C(Time, X)) inferInstance hπ.1 hπ.2
have ht := localized_excess_zero_iff μ ((π : Measure C(Time, X)).map (fun γ => γ midpointTime))
(B := W) (C := C) hs hC
exact ht.mp hz
end Attainment
end RegularPointCoordinates
end OAI
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