| import OAI.Analysis.CircleDomains.Probability.MarkedHitFusion |
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| namespace OAI |
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|
| /-! The fusion theorem applied to a closed dust subset. Its tests are ordinary |
| ambient measures supported on the dust, exactly as in the modulus extraction. -/ |
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|
| open Set Function MeasureTheory |
| open scoped Topology ENNReal |
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| noncomputable section |
| namespace CircleDomainRigidity |
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|
| variable {X Z : Type*} [MetricSpace X] [CompactSpace X] [SecondCountableTopology X] |
| [MeasurableSpace X] [BorelSpace X] |
| [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] |
| [MetricSpace Z] [SecondCountableTopology Z] [CompleteSpace Z] |
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| theorem exists_hit_norm_mixture_on_closed_subset |
| (B : Set X) (hB : IsClosed B) |
| (law : Z → ProbabilityMeasure C(PathTime, X)) (hlaw : Continuous law) |
| {C : ℝ} (hC : 0 ≤ C) |
| (htest : ∀ D : Set X, D.Countable → D ⊆ B → ∀ ν : Measure X, |
| IsFiniteMeasure ν → NullSingletonClass ν → ν Bᶜ = 0 → |
| ∃ z, PassesHitTests C (law z) D ν) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (P : ProbabilityMeasure (ℕ → ℕ)) (e : C(ℕ → ℕ, Z)), |
| let μ := probabilityMixture P (law ∘ e) |
| (measurable_of_continuous_probability (hlaw.comp e.continuous)) |
| (∀ F : Finset B, ∑ x ∈ F, pathHitReal μ x.val ^ 2 ≤ (C + ε) ^ 2) ∧ |
| {x : X | x ∈ B ∧ 0 < pathHitProbability μ x}.Countable := by |
| classical |
| let : CompactSpace B := isCompact_iff_compactSpace.mp hB.isCompact |
| let ι : C(B, X) := ⟨Subtype.val, continuous_subtype_val⟩ |
| have htest' : ∀ D : Set B, D.Countable → ∀ ν : Measure B, |
| IsFiniteMeasure ν → NullSingletonClass ν → |
| ∃ z, MarkedHit.PassesHitTests ι C (law z) D ν := by |
| intro D hD ν hν hnull |
| let : IsFiniteMeasure ν := hν |
| let : NullSingletonClass ν := hnull |
| let ρ : Measure X := ν.map (Subtype.val : B → X) |
| have hρ : IsFiniteMeasure ρ := inferInstance |
| have hρnull : NullSingletonClass ρ := by |
| constructor |
| intro x |
| rw [show ρ = ν.map (Subtype.val : B → X) from rfl, |
| Measure.map_apply continuous_subtype_val.measurable (measurableSet_singleton x)] |
| by_cases hx : x ∈ B |
| · have he : (Subtype.val : B → X) ⁻¹' {x} = {⟨x, hx⟩} := by |
| ext y |
| simp only [mem_preimage, mem_singleton_iff, Subtype.ext_iff] |
| rw [he] |
| exact measure_singleton _ |
| · have he : (Subtype.val : B → X) ⁻¹' {x} = ∅ := by |
| apply eq_empty_iff_forall_notMem.mpr |
| intro y hy |
| exact hx ((mem_singleton_iff.mp hy) ▸ y.property) |
| rw [he, measure_empty] |
| have hρB : ρ Bᶜ = 0 := by |
| rw [show ρ = ν.map (Subtype.val : B → X) from rfl, |
| Measure.map_apply continuous_subtype_val.measurable hB.measurableSet.compl] |
| simp |
| obtain ⟨z, hz⟩ := htest (Subtype.val '' D) (hD.image _) (by |
| rintro x ⟨y, _, rfl⟩ |
| exact y.property) ρ hρ hρnull hρB |
| refine ⟨z, ?_, ?_⟩ |
| · intro F hF |
| have hb := hz.1 (F.image (Subtype.val : B → X)) (by |
| intro x hx |
| obtain ⟨y, hy, rfl⟩ := Finset.mem_image.mp hx |
| exact mem_image_of_mem _ (hF hy)) |
| simpa only [Finset.sum_image (fun _ _ _ _ h => Subtype.val_injective h), |
| MarkedHit.pathHitReal, MarkedHit.pathHitProbability, pathHitReal, ι, |
| ContinuousMap.coe_mk] using hb |
| · change ∫⁻ x, pathHitProbability (law z) (x : X) ∂ν = 0 |
| rw [← lintegral_map (measurable_pathHitProbability_point (law z)) |
| continuous_subtype_val.measurable] |
| exact hz.2 |
| obtain ⟨P, e, hb⟩ := MarkedHit.exists_hit_norm_mixture_of_countable_tests ι hlaw hC htest' hε |
| refine ⟨P, e, hb, ?_⟩ |
| have hcount := hb.countable_positive_support |
| have heq : Subtype.val '' {x : B | 0 < MarkedHit.pathHitProbability ι |
| (probabilityMixture P (law ∘ e) |
| (measurable_of_continuous_probability (hlaw.comp e.continuous))) x} = |
| {x : X | x ∈ B ∧ 0 < pathHitProbability |
| (probabilityMixture P (law ∘ e) |
| (measurable_of_continuous_probability (hlaw.comp e.continuous))) x} := by |
| ext x |
| constructor |
| · rintro ⟨y, hy, rfl⟩ |
| exact ⟨y.property, hy⟩ |
| · rintro ⟨hx, hh⟩ |
| exact ⟨⟨x, hx⟩, hh, rfl⟩ |
| rw [← heq] |
| exact hcount.image _ |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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