| 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 |
| |
