File size: 3,234 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 | import OAI.Analysis.CircleDomains.Probability.CodeLawCandidates
import OAI.Analysis.CircleDomains.Probability.DensityMassBound
import OAI.Analysis.CircleDomains.Probability.CountableVisitNorm
namespace OAI
/-! The quantitative outputs of countable modulus extraction are members of
one fixed candidate family, independent of the prescribed countable set. -/
open Set MeasureTheory
open scoped ENNReal
noncomputable section
namespace CircleDomainRigidity
variable {X Y S : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [BorelSpace X] [SecondCountableTopology X]
[MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
omit [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in
theorem meanVisitCount_codePathLaw_eq_integral (d : PathCodeData X Y)
(P : ProbabilityMeasure (BasePathCode d)) (a : S → X) (s : S) :
meanVisitCount (codePathLaw d P) (a s) =
∫⁻ Γ, codeVisitWeight d a s Γ ∂(P : Measure (BasePathCode d)) := by
have hv : Measurable (fun γ : C(PathTime, X) => visitCount γ (a s)) :=
measurable_visitCount.comp (measurable_id.prodMk measurable_const)
unfold meanVisitCount codePathLaw
rw [ProbabilityMeasure.toMeasure_map, lintegral_map hv (measurable_codePath d)]
rfl
omit [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in
theorem countableSquareLawBounds_of_visit_norm (d : PathCodeData X Y)
(P : ProbabilityMeasure (BasePathCode d)) (a : S → X) {C : ℝ≥0∞}
(h : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C) :
CountableSquareLawBounds (codeVisitWeight d a) (C ^ 2) P := by
intro F
simp_rw [← meanVisitCount_codePathLaw_eq_integral]
have he : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ^ 2 =
∑' s, meanVisitCount (codePathLaw d P) (a s) ^ 2 := by
rw [countingSquareNorm, ← ENNReal.rpow_two, ← ENNReal.rpow_mul]
norm_num
exact (ENNReal.sum_le_tsum F).trans (he ▸ pow_le_pow_left' h 2)
omit [SecondCountableTopology Y] [MeasurableSpace C(PathTime, Y)]
[BorelSpace C(PathTime, Y)] in
theorem codeLawCandidate_of_extracted_bounds (d : PathCodeData X Y)
(a : S → X) (μ : Measure Y) (τ : Measure LevelTuple)
(P : ProbabilityMeasure (BasePathCode d))
{g : Y → ℝ≥0∞} {q : LevelTuple → ℝ≥0∞} (hg : Measurable g) (hq : Measurable q)
(hocc : meanCodeOccupation d P = μ.withDensity g)
(htuple : meanCodeTuple d P = τ.withDensity q)
{E C : ℝ≥0∞} (hgE : (∫⁻ y, g y ^ 2 ∂μ) ≤ E)
(hqE : (∫⁻ t, q t ^ 2 ∂τ) ≤ E)
(hv : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C) :
CodeLawCandidate d a μ τ (densityMassBound μ E) (densityMassBound τ E) E (C ^ 2) P :=
⟨boundedDensityBudget_of_square_bound μ _ hg hocc hgE,
countableSquareLawBounds_of_visit_norm d P a hv,
boundedDensityBudget_of_square_bound τ _ hq htuple hqE⟩
end CircleDomainRigidity
end
end OAI
|