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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 | import OAI.Analysis.CircleDomains.Probability.CodeLawCandidateBounds
import OAI.Analysis.CircleDomains.Probability.SlotLawMeasures
namespace OAI
/-! Unpack the fused candidate's quantitative estimates and slot laws. -/
open Set MeasureTheory Filter
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]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
omit [MeasurableSpace Y] [BorelSpace Y] in
theorem countableSquareLawBounds_visit_norm (d : PathCodeData X Y)
(P : ProbabilityMeasure (BasePathCode d)) (a : S → X) {C : ℝ≥0∞}
(h : CountableSquareLawBounds (codeVisitWeight d a) (C ^ 2) P) :
countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C := by
have hs : (∑' s, meanVisitCount (codePathLaw d P) (a s) ^ 2) ≤ C ^ 2 := by
rw [ENNReal.tsum_eq_iSup_sum]
apply iSup_le
intro F
simpa only [meanVisitCount_codePathLaw_eq_integral] using h F
calc
_ ≤ (C ^ 2) ^ (1 / 2 : ℝ) := ENNReal.rpow_le_rpow hs (by norm_num)
_ = C := by rw [← ENNReal.rpow_two, ← ENNReal.rpow_mul]; norm_num
omit [BorelSpace X] [SecondCountableTopology X] [BorelSpace C(PathTime, X)] in
theorem CodeLawCandidate.slot_absolutelyContinuous
(d : PathCodeData X Y) (a : S → X) (μ : Measure Y) (τ : Measure LevelTuple)
(M N E V : ℝ≥0∞) (P : ProbabilityMeasure (BasePathCode d))
(h : CodeLawCandidate d a μ τ M N E V P) (k : ℕ) : slotLaw d P k ≪ τ := by
obtain ⟨q, _, hq, _⟩ := h.2.2.2
exact slotLaw_absolutelyContinuous d P τ (hq ▸ withDensity_absolutelyContinuous τ q) k
omit [MeasurableSpace Y] [BorelSpace Y] in
theorem ae_finite_prescribed_visits [Countable S]
(d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) (a : S → X)
{V : ℝ≥0∞} (hV : V < ∞)
(h : CountableSquareLawBounds (codeVisitWeight d a) V P) :
∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ s, visitCount (codePath d Γ) (a s) < ∞ := by
classical
apply ae_all_iff.mpr
intro s
have hb : (∫⁻ Γ, codeVisitWeight d a s Γ ∂(P : Measure (BasePathCode d))) ^ 2 < ∞ := by
simpa using (h {s}).trans_lt hV
apply ae_lt_top (measurable_codeVisitWeight d a s)
intro hh
rw [hh] at hb
simp at hb
end CircleDomainRigidity
end
end OAI
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