| import OAI.Analysis.CircleDomains.Selection.TaggedPointContacts | |
| namespace OAI | |
| /-! Fubini for complete path images. At free points, tuple absolute | |
| continuity reduces a path hit to a residual hit; residual nullity then | |
| annihilates the integrated whole-image hit probability. -/ | |
| open Set MeasureTheory Filter | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] | |
| [MeasurableSpace X] [MeasurableSpace C(PathTime, X)] | |
| def codeHitGraph (d : PathCodeData X Y) : Set (BasePathCode d × X) := | |
| {q | q.2 ∈ Set.range q.1.val.1} | |
| def codeResidualGraph (d : PathCodeData X Y) : Set (BasePathCode d × X) := | |
| {q | q.2 ∈ residualImage d.dust q.1.val} | |
| theorem isClosed_codeHitGraph (d : PathCodeData X Y) : IsClosed (codeHitGraph d) := | |
| isClosed_pathHitSet.preimage | |
| (((continuous_subtype_val.comp continuous_fst).fst).prodMk continuous_snd) | |
| theorem isClosed_codeResidualGraph (d : PathCodeData X Y) (hB : IsClosed d.dust) : | |
| IsClosed (codeResidualGraph d) := | |
| (isClosed_residualImageGraph hB).preimage | |
| ((continuous_subtype_val.comp continuous_fst).prodMk continuous_snd) | |
| def codeHitProbability (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) | |
| (x : X) : ℝ≥0∞ := | |
| (P : Measure (BasePathCode d)) {Γ | x ∈ Set.range Γ.val.1} | |
| def codeResidualProbability (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) | |
| (x : X) : ℝ≥0∞ := | |
| (P : Measure (BasePathCode d)) {Γ | x ∈ residualImage d.dust Γ.val} | |
| variable [BorelSpace X] [BorelSpace C(PathTime, X)] [SecondCountableTopology X] | |
| theorem measurable_codeHitProbability (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) : Measurable (codeHitProbability d P) := | |
| measurable_measure_prodMk_right (isClosed_codeHitGraph d).measurableSet | |
| theorem measurable_codeResidualProbability (d : PathCodeData X Y) (hB : IsClosed d.dust) | |
| (P : ProbabilityMeasure (BasePathCode d)) : Measurable (codeResidualProbability d P) := | |
| measurable_measure_prodMk_right (isClosed_codeResidualGraph d hB).measurableSet | |
| theorem lintegral_codeResidualProbability (d : PathCodeData X Y) (hB : IsClosed d.dust) | |
| (P : ProbabilityMeasure (BasePathCode d)) (ν : Measure X) [SFinite ν] : | |
| ∫⁻ x, codeResidualProbability d P x ∂ν = | |
| ∫⁻ Γ : BasePathCode d, ν (residualImage d.dust Γ.val) ∂(P : Measure (BasePathCode d)) := by | |
| have hm := (isClosed_codeResidualGraph d hB).measurableSet | |
| exact (Measure.prod_apply_symm hm).symm.trans (Measure.prod_apply hm) | |
| omit [BorelSpace X] [BorelSpace C(PathTime, X)] [SecondCountableTopology X] in | |
| theorem codeHitProbability_eq_residual (d : PathCodeData X Y) | |
| (hH : ∀ i, IsClosed (d.levels i)) (P : ProbabilityMeasure (BasePathCode d)) | |
| (ω : TupleType → ℝ≥0∞) (hτ : meanCodeTuple d P ≪ tupleReference ω) | |
| {x : X} (hx : x ∈ d.dust) | |
| (hfree : ∀ i, volume {t : UnitAddCircle | (t, x) ∈ d.levels i} = 0) : | |
| codeHitProbability d P x = codeResidualProbability d P x := by | |
| apply measure_congr | |
| filter_upwards [ae_path_hit_iff_residual d hH P ω hτ hx hfree] with Γ hΓ | |
| exact propext hΓ | |
| theorem lintegral_codeHitProbability_zero (d : PathCodeData X Y) (hB : IsClosed d.dust) | |
| (hH : ∀ i, IsClosed (d.levels i)) (P : ProbabilityMeasure (BasePathCode d)) | |
| (ω : TupleType → ℝ≥0∞) (hτ : meanCodeTuple d P ≪ tupleReference ω) | |
| (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 := by | |
| calc | |
| _ = ∫⁻ x, codeResidualProbability d P x ∂ν := | |
| lintegral_congr_ae (hfree.mono fun x hx => | |
| codeHitProbability_eq_residual d hH P ω hτ hx.1 hx.2) | |
| _ = ∫⁻ Γ : BasePathCode d, ν (residualImage d.dust Γ.val) | |
| ∂(P : Measure (BasePathCode d)) := lintegral_codeResidualProbability d hB P ν | |
| _ = 0 := (lintegral_eq_zero_iff | |
| ((measurable_residualImageMass hB ν).comp measurable_subtype_coe)).mpr hres | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |