openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /ClosedProbabilitySupport.lean
| import Mathlib.MeasureTheory.Measure.Portmanteau | |
| import Mathlib.Topology.Semicontinuity.Basic | |
| namespace OAI | |
| /-! Support in a closed set is retained by weak limits of probability laws. -/ | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma isClosed_probability_supported_closed | |
| {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) : | |
| IsClosed {P : ProbabilityMeasure X | ∀ᵐ x ∂(P : Measure X), x ∈ F} := by | |
| have hsc : UpperSemicontinuous (fun P : ProbabilityMeasure X => (P : Measure X) F) := | |
| upperSemicontinuous_iff_limsup_le.mpr (fun P => | |
| ProbabilityMeasure.limsup_measure_closed_le_of_tendsto tendsto_id hF) | |
| have he : {P : ProbabilityMeasure X | ∀ᵐ x ∂(P : Measure X), x ∈ F} = | |
| {P : ProbabilityMeasure X | (1 : ℝ≥0∞) ≤ (P : Measure X) F} := by | |
| ext P | |
| change (∀ᵐ x ∂(P : Measure X), x ∈ F) ↔ (1 : ℝ≥0∞) ≤ (P : Measure X) F | |
| constructor | |
| · intro h | |
| have heq : F =ᵐ[(P : Measure X)] univ := h.mono (fun x hx => by simp [hx]) | |
| rw [measure_congr heq,measure_univ] | |
| · intro h | |
| have hm : (P : Measure X) F = 1 := le_antisymm ((measure_mono (subset_univ F)).trans_eq measure_univ) h | |
| have hc : (P : Measure X) Fᶜ = 0 := by | |
| rw [measure_compl hF.measurableSet (measure_ne_top _ _),hm,measure_univ,tsub_self] | |
| apply ae_iff.mpr | |
| change (P : Measure X) Fᶜ = 0 | |
| exact hc | |
| rw [he] | |
| exact hsc.isClosed_preimage 1 | |
| lemma probability_supported_closed_of_tendsto | |
| {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [HasOuterApproxClosed X] {P : ℕ → ProbabilityMeasure X} {Q : ProbabilityMeasure X} | |
| (hlim : Tendsto P atTop (𝓝 Q)) {F : Set X} (hF : IsClosed F) | |
| (hsupp : ∀ᶠ n in atTop, ∀ᵐ x ∂(P n : Measure X), x ∈ F) : | |
| ∀ᵐ x ∂(Q : Measure X), x ∈ F := | |
| (isClosed_probability_supported_closed hF).mem_of_tendsto hlim hsupp | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |