openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CodeLawAbsoluteContinuity.lean
| import OAI.Analysis.CircleDomains.Modulus.CostSummands | |
| import OAI.Analysis.CircleDomains.Probability.CodeHitMeasures | |
| namespace OAI | |
| /-! A density for the total coding barycenter gives absolute continuity | |
| on each cost summand, including every tuple slot and the whole-image | |
| atomless tests. -/ | |
| open Set MeasureTheory Filter | |
| open scoped ENNReal | |
| 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] | |
| theorem codeLaw_components_absolutelyContinuous (d : PathCodeData X Y) | |
| (hU : IsOpen d.quotientDomain) (p : D → X) (c : D → ℝ≥0∞) | |
| (P : ProbabilityMeasure (BasePathCode d)) (μ : Measure Y) (τ : Measure LevelTuple) | |
| (q : PathCostSpace Y D → ℝ≥0∞) | |
| (hbar : meanCodeCost d p c P = (pathCostReference μ c τ).withDensity q) : | |
| meanCodeOccupation d P ≪ μ ∧ meanCodeAtomic d p c P ≪ atomicReference c ∧ | |
| meanCodeTuple d P ≪ τ := by | |
| have hac := withDensity_absolutelyContinuous (pathCostReference μ c τ) q | |
| rw [← hbar, meanCodeCost_eq d hU p c P] at hac | |
| exact sumCostMeasures_absolutelyContinuous hac | |
| theorem codeLaw_slot_absolutelyContinuous (d : PathCodeData X Y) | |
| (hU : IsOpen d.quotientDomain) (p : D → X) (c : D → ℝ≥0∞) | |
| (P : ProbabilityMeasure (BasePathCode d)) (μ : Measure Y) (τ : Measure LevelTuple) | |
| (q : PathCostSpace Y D → ℝ≥0∞) | |
| (hbar : meanCodeCost d p c P = (pathCostReference μ c τ).withDensity q) (k : ℕ) : | |
| slotLaw d P k ≪ τ := | |
| slotLaw_absolutelyContinuous d P τ | |
| (codeLaw_components_absolutelyContinuous d hU p c P μ τ q hbar).2.2 k | |
| theorem codeLaw_whole_image_test (d : PathCodeData X Y) | |
| (hU : IsOpen d.quotientDomain) (hB : IsClosed d.dust) | |
| (hH : ∀ i, IsClosed (d.levels i)) (p : D → X) (c : D → ℝ≥0∞) | |
| (P : ProbabilityMeasure (BasePathCode d)) (μ : Measure Y) (ω : TupleType → ℝ≥0∞) | |
| (q : PathCostSpace Y D → ℝ≥0∞) | |
| (hbar : meanCodeCost d p c P = (pathCostReference μ c (tupleReference ω)).withDensity q) | |
| (ν : Measure X) [SFinite ν] | |
| (hfree : ∀ᵐ x ∂ν, x ∈ d.dust ∧ | |
| ∀ i, volume {t : UnitAddCircle | (t, x) ∈ d.levels i} = 0) | |
| (hres : ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ν (residualImage d.dust Γ.val) = 0) : | |
| ∫⁻ x, codeHitProbability d P x ∂ν = 0 := | |
| lintegral_codeHitProbability_zero d hB hH P ω | |
| (codeLaw_components_absolutelyContinuous d hU p c P μ (tupleReference ω) q hbar).2.2 | |
| ν hfree hres | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |