| import OAI.Analysis.CircleDomains.Topology.TerminalArcSet |
| import Mathlib.Probability.ConditionalProbability |
|
|
| namespace OAI |
|
|
| /-! Conditioning the terminal-arc law on a compact family of mass greater |
| than one half loses at most a factor two in every measure bound. -/ |
|
|
| open Set Function Filter MeasureTheory ProbabilityTheory |
| open scoped ENNReal Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| section Restriction |
| variable {Ω : Type*} [MeasurableSpace Ω] |
|
|
| def restrictProbability (μ : ProbabilityMeasure Ω) (K : Set Ω) |
| (hK : (μ : Measure Ω) K ≠ 0) : ProbabilityMeasure Ω := |
| ⟨cond (μ : Measure Ω) K, cond_isProbabilityMeasure hK⟩ |
|
|
| theorem restrictProbability_le_two (μ : ProbabilityMeasure Ω) (K : Set Ω) |
| (hK : (1 / 2 : ℝ≥0∞) ≤ (μ : Measure Ω) K) : |
| (restrictProbability μ K (ne_of_gt ((by norm_num : (0 : ℝ≥0∞) < 1/2).trans_le hK)) : Measure Ω) ≤ |
| (2 : ℝ≥0∞) • (μ : Measure Ω) := by |
| apply Measure.le_iff.mpr |
| intro S hS |
| change cond (μ : Measure Ω) K S ≤ _ |
| rw [cond_apply' hS, Measure.smul_apply, smul_eq_mul] |
| apply mul_le_mul' ?_ (measure_mono inter_subset_right) |
| have hi := ENNReal.inv_le_inv.mpr hK |
| norm_num at hi ⊢ |
| exact hi |
|
|
| theorem restrictProbability_ae_mem (μ : ProbabilityMeasure Ω) (K : Set Ω) |
| (hK : (μ : Measure Ω) K ≠ 0) (hm : MeasurableSet K) : |
| ∀ᵐ x ∂(restrictProbability μ K hK : Measure Ω), x ∈ K := ae_cond_mem hm |
|
|
| end Restriction |
|
|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] |
|
|
| theorem exists_compact_terminal_arc_law |
| (μ : ProbabilityMeasure C(PathTime, Sphere)) (V : Set Sphere) (hV : IsOpen V) |
| (b : Sphere) (hμ : ∀ᵐ α : C(PathTime, Sphere) ∂(μ : Measure C(PathTime, Sphere)), |
| α ∈ terminalArcSet V b) : |
| ∃ (K : Set C(PathTime, Sphere)) (η : ProbabilityMeasure C(PathTime, Sphere)), |
| IsCompact K ∧ K ⊆ terminalArcSet V b ∧ |
| (∀ᵐ α ∂(η : Measure C(PathTime, Sphere)), α ∈ K) ∧ |
| (η : Measure C(PathTime, Sphere)) ≤ (2 : ℝ≥0∞) • (μ : Measure C(PathTime, Sphere)) := by |
| obtain ⟨K, hKS, hK, hmass⟩ := exists_compact_terminal_arc_core μ V hV b hμ |
| have hpos : (μ : Measure C(PathTime, Sphere)) K ≠ 0 := |
| ne_of_gt ((by norm_num : (0 : ℝ≥0∞) < 1/2).trans hmass) |
| exact ⟨K, restrictProbability μ K hpos, hK, hKS, |
| restrictProbability_ae_mem μ K hpos hK.measurableSet, |
| restrictProbability_le_two μ K hmass.le⟩ |
|
|
| theorem exists_compact_selected_arc_law |
| (hselect : BorelPathRelationSelection) |
| (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (μ : ProbabilityMeasure C(PathTime, Sphere)) (V : Set Sphere) (hV : IsOpen V) |
| (b : Sphere) (hμ : ∀ᵐ γ : C(PathTime, Sphere) ∂(μ : Measure C(PathTime, Sphere)), |
| γ ⊥ ∉ V ∧ γ ⊤ ∈ V ∧ γ ⊤ = b) : |
| ∃ (K : Set C(PathTime, Sphere)) (η : ProbabilityMeasure C(PathTime, Sphere)), |
| IsCompact K ∧ K ⊆ terminalArcSet V b ∧ |
| (∀ᵐ α ∂(η : Measure C(PathTime, Sphere)), α ∈ K) ∧ |
| (∀ x, pathHitProbability η x ≤ 2 * pathHitProbability μ x) ∧ |
| (∀ x ∉ closure V, pathHitProbability η x = 0) := by |
| obtain ⟨f, hfm, hf⟩ := exists_measurable_terminal_arc_selector hselect harcs μ V hV |
| obtain ⟨K, η, hK, hKS, hη, hbound⟩ := exists_compact_terminal_arc_law (μ.map f) V hV b |
| (terminalArcLaw_mem μ V hV b f hfm hf hμ) |
| have hends : ∀ᵐ γ : C(PathTime, Sphere) ∂(μ : Measure C(PathTime, Sphere)), |
| γ ⊥ ∉ V ∧ γ ⊤ ∈ V := hμ.mono fun _ h => ⟨h.1, h.2.1⟩ |
| refine ⟨K, η, hK, hKS, hη, ?_, ?_⟩ |
| · intro x |
| exact (hbound _).trans (mul_le_mul' le_rfl |
| (terminalArcLaw_hit_le μ V hV f hfm hf hends x)) |
| · intro x hx |
| apply le_antisymm ?_ bot_le |
| have h := hbound {α | ∃ t : PathTime, α t = x} |
| change pathHitProbability η x ≤ 2 * pathHitProbability (μ.map f) x at h |
| rw [terminalArcLaw_hit_eq_zero μ V hV f hfm hf hends hx, mul_zero] at h |
| exact h |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
