File size: 3,494 Bytes
9655094 | 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 69 70 71 72 73 74 75 76 | import OAI.Analysis.CircleDomains.Probability.PublishedMeasureDuality
import OAI.Analysis.CircleDomains.Probability.ConstrainedCodeLaws
import OAI.Analysis.CircleDomains.Modulus.RetaggedCost
import OAI.Analysis.CircleDomains.Probability.CountableVisitNorm
import OAI.Analysis.CircleDomains.Selection.CodeBarycenterDensities
import OAI.Analysis.CircleDomains.Selection.FreeLevelSections
namespace OAI
/-! Measure-family duality and probability lifting produce a law on the
constrained codes from a positive modulus bound. -/
open Set MeasureTheory Filter
open scoped ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
variable {X Y D : Type*} [MetricSpace X] [MetricSpace Y] [TopologicalSpace D]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[MeasurableSpace D] [BorelSpace D]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[SecondCountableTopology X] [Countable D]
omit [BorelSpace X] [BorelSpace D] [BorelSpace C(PathTime, X)]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[SecondCountableTopology X] [Countable D] in
theorem costAdmissible_iff_constrained (d : PathCodeData X Y)
(p : D → X) (c : D → ℝ≥0∞) (ν : Measure X)
(h : PathCostSpace Y D → ℝ≥0∞) :
CostAdmissible d p c ν h ↔
∀ Γ : ConstrainedCostCode d p c ν,
1 ≤ ∫⁻ z, h z ∂(constrainedCodeMeasure d p c ν Γ : Measure (PathCostSpace Y D)) := by
constructor
· intro hadm Γ
exact hadm Γ.val.val Γ.val.property Γ.property
· intro hadm Γ hfinite hres
exact hadm ⟨⟨Γ, hfinite⟩, hres⟩
variable [PolishSpace Y] [PolishSpace D]
theorem positive_modulus_code_law (d : PathCodeData X Y)
(hU : IsOpen d.quotientDomain) (hB : IsClosed d.dust)
(p : D → X) (c : D → ℝ≥0∞) (ν : Measure X) [SFinite ν]
[StandardBorelSpace (ConstrainedCostCode d p c ν)]
(μ : Measure Y) [IsFiniteMeasure μ] (τ : Measure LevelTuple) [IsFiniteMeasure τ]
(hc : (∑' s, c s ^ 2) < ∞)
(hdual : MeasureImageDuality (ConstrainedCostCode d p c ν) (PathCostSpace Y D))
(hlift : MeasurePlanLifting (ConstrainedCostCode d p c ν) (PathCostSpace Y D))
(r : ℝ≥0) (hr : 0 < r)
(hmod : (r : ℝ≥0∞) < measureFamilyModulus (ConstrainedCostCode d p c ν)
(PathCostSpace Y D) (pathCostReference μ c τ) (constrainedCodeMeasure d p c ν)) :
∃ (P : ProbabilityMeasure (BasePathCode d)) (q : PathCostSpace Y D → ℝ≥0∞),
Measurable q ∧
meanCodeCost d p c P = (pathCostReference μ c τ).withDensity q ∧
(∫⁻ z, q z ^ 2 ∂pathCostReference μ c τ) ≤ (r : ℝ≥0∞)⁻¹ ∧
(∀ᵐ Γ ∂(P : Measure (BasePathCode d)), pathCostMeasure d p c Γ univ < ∞) ∧
(∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ν (residualImage d.dust Γ.val) = 0) := by
let : IsFiniteMeasure (pathCostReference μ c τ) := finite_pathCostReference μ c hc τ
obtain ⟨P, q, hq, hbar, he⟩ := indexed_measure_duality _ _ hdual hlift
(constrainedCodeMeasure d p c ν) (measurable_constrainedCodeMeasure d p c ν hU)
(pathCostReference μ c τ) r hr hmod
refine ⟨constrainedCodeLaw d p c ν P, q, hq, ?_, he,
constrainedCodeLaw_finite d hU p c ν P, constrainedCodeLaw_residual_null d p c ν hB P⟩
ext E hE
rw [mean_constrainedCodeLaw d hU p c ν P hE]
exact hbar E hE
end CircleDomainRigidity
end
end OAI
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