| import OAI.Analysis.CircleDomains.Rigidity.CodeFreeTrace |
|
|
| namespace OAI |
|
|
| /-! Combining the actual residual constraints with null incidence sets |
| on the open tags needs no globally chosen circle trace. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped Topology ENNReal |
|
|
| namespace CircleDomainRigidity |
|
|
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [BorelSpace X] [SecondCountableTopology X] |
| [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] |
|
|
| theorem ae_code_free_level_incidence_null_of_tags |
| (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) (i : ℕ) |
| (Free : Set X) (hFree : Free ⊆ d.dust) |
| (hm : (support (d.levelWeights i)).Countable) |
| (hweight : countingSquareNorm (fun x : support (d.levelWeights i) => |
| d.levelWeights i x.val) < ∞) |
| (hvisits : countingSquareNorm (fun x : support (d.levelWeights i) => |
| meanVisitCount (codePathLaw d P) x.val) < ∞) |
| (hcover : ∀ x ∈ support (d.levelWeights i), ∀ᶠ n in atTop, x ∈ d.flagDeletions n) |
| (hdelete : ∀ n, Disjoint Free (d.flagDeletions n : Set X)) |
| (htags : ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ k, |
| volume {θ | ∃ t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k)), |
| Γ.val.1 t ∈ Free ∧ (θ, Γ.val.1 t) ∈ d.levels i} = 0) : |
| ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), |
| volume {θ | ∃ x ∈ range Γ.val.1 ∩ Free, (θ, x) ∈ d.levels i} = 0 := by |
| filter_upwards [ae_code_level_envelope_null d P i hm hweight hvisits hcover, htags] |
| with Γ hres htag |
| apply measure_mono_null (t := |
| (⋂ n, {θ | (Γ.val, θ) ∈ residualIncidence d.dust univ (d.levels i) (d.flagDeletions n)}) ∪ |
| ⋃ k, {θ | ∃ t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k)), |
| Γ.val.1 t ∈ Free ∧ (θ, Γ.val.1 t) ∈ d.levels i}) _ |
| (measure_union_null hres (measure_iUnion_null htag)) |
| rintro θ ⟨x, ⟨⟨t, rfl⟩, hx⟩, hθ⟩ |
| by_cases ht : ∃ k, t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k)) |
| · obtain ⟨k, hk⟩ := ht |
| exact Or.inr (mem_iUnion.mpr ⟨k, t, hk, hx, hθ⟩) |
| · apply Or.inl |
| apply mem_iInter.mpr |
| intro n |
| refine ⟨t, ?_, mem_univ _, hθ, fun hp => disjoint_left.mp (hdelete n) hx hp⟩ |
| rw [residualTimes_eq] |
| exact ⟨hFree hx, fun h => ht (mem_iUnion.mp h)⟩ |
|
|
| end CircleDomainRigidity |
|
|
| end OAI |
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|
