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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 | import OAI.Analysis.CircleDomains.Topology.SphereLevels
import OAI.Analysis.CircleDomains.Probability.PhysicalOccupationCode
import OAI.Analysis.CircleDomains.Probability.CodePathLaws
import OAI.Analysis.CircleDomains.Probability.CodeLawMeasures
namespace OAI
/-! Forgetting tags preserves the actual mean physical occupation. -/
open Set Function Filter MeasureTheory
open scoped ENNReal Topology
noncomputable section
namespace CircleDomainRigidity
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)]
theorem meanPhysicalOccupation_codePathLaw (d : PathCodeData Sphere Sphere)
(hU : IsOpen d.quotientDomain) (P : ProbabilityMeasure (BasePathCode d)) :
meanPhysicalOccupation d.domainEquiv d.basePoint (codePathLaw d P) =
meanCodeOccupation d P := by
apply Measure.ext
intro S hS
have hm : Measurable (fun γ => physicalOccupationCode d.domainEquiv d.basePoint γ S) :=
(Measure.measurable_coe hS).comp (measurable_physicalOccupationCode d.domainEquiv hU d.basePoint)
rw [meanPhysicalOccupation_apply _ hU _ _ hS, meanCodeOccupation_apply d hU P hS,
codePathLaw, ProbabilityMeasure.toMeasure_map, lintegral_map hm (measurable_codePath d)]
apply lintegral_congr
intro Γ
exact congrArg (fun ν : Measure Sphere => ν S)
(physicalOccupationCode_eq d.domainEquiv d.basePoint (baseCodePhysicalPath d Γ))
theorem codePathLaw_collar_endpoints (d : PathCodeData Sphere Sphere)
(hU : IsOpen d.quotientDomain) (P : ProbabilityMeasure (BasePathCode d))
(V : Set Sphere) (hV : IsOpen V) (hanchor : Disjoint d.anchor V) (hb : d.target ∈ V) :
∀ᵐ γ : C(PathTime, Sphere) ∂(codePathLaw d P : Measure C(PathTime, Sphere)),
γ ⊥ ∉ V ∧ γ ⊤ ∈ V ∧ γ ⊤ = d.target ∧ physicalDomainLength d.domainEquiv d.basePoint γ < ∞ := by
change ∀ᵐ γ ∂(P : Measure (BasePathCode d)).map (codePath d), _
apply (ae_map_iff (measurable_codePath d).aemeasurable ?_).mpr
· exact Eventually.of_forall fun Γ =>
⟨fun hv => disjoint_left.mp hanchor Γ.property.2.1.1 hv,
by change Γ.val.1 ⊤ ∈ V; rw [Γ.property.2.1.2.1]; exact hb,
Γ.property.2.1.2.1, Γ.property.1.1⟩
· exact (((continuous_eval_const ⊥).measurable hV.measurableSet).compl).inter
(((continuous_eval_const ⊤).measurable hV.measurableSet).inter
((measurableSet_eq_fun (continuous_eval_const ⊤).measurable measurable_const).inter
(measurableSet_finitePhysicalDomainLength d.domainEquiv hU d.basePoint)))
end CircleDomainRigidity
end
end OAI
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