File size: 2,837 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 | 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
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