openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CompactMeasureContinuity.lean
| import OAI.Analysis.CircleDomains.Topology.NestedCompactTestPlan | |
| import OAI.Analysis.CircleDomains.Probability.CompactPlanTestDensity | |
| import OAI.Analysis.CircleDomains.Sobolev.MeasurePlanWeakDuality | |
| namespace OAI | |
| /-! Continuity of pointwise modulus on decreasing compact measure families | |
| avoiding zero. The limiting probability plan supplies the lower bound. -/ | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal BoundedContinuousFunction | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma compact_intersection_modulus_lower | |
| {Z : Type*} [TopologicalSpace Z] [PolishSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] | |
| (σ : Measure Z) [IsFiniteMeasure σ] | |
| (T : ℕ → Set (FiniteMeasure Z)) (hT : ∀ n, IsCompact (T n)) | |
| (hne : ∀ n, (T n).Nonempty) (hanti : Antitone T) | |
| (hz : (0 : FiniteMeasure Z) ∉ T 0) {c : ℝ≥0} (hc : 0 < c) | |
| (hm : ∀ n, (c : ℝ≥0∞) < measureFamilyModulus (T n) Z σ (fun v => v.val)) : | |
| (c : ℝ≥0∞) ≤ measureFamilyModulus (⋂ n, T n) Z σ (fun v => v.val) := by | |
| let : CompactSpace (T 0) := isCompact_iff_compactSpace.mp (hT 0) | |
| have hcr : (0 : ℝ) < c := hc | |
| have hsq : Real.sqrt (c : ℝ)^2 = c := Real.sq_sqrt c.coe_nonneg | |
| obtain ⟨P,hPs,hPt⟩ := exists_nested_compact_test_plan σ T hT hne hanti hz | |
| (Real.sqrt_pos.mpr hcr) (by simpa only [hsq,ENNReal.ofReal_coe_nnreal] using hm) | |
| obtain ⟨q,hq,hbar,he⟩ := compact_plan_density_of_test_bound σ (hT 0) P | |
| (Real.sqrt_pos.mpr hcr) hPt | |
| have he' : (∫⁻ z, q z^2 ∂σ) ≤ (c : ℝ≥0∞)⁻¹ := by | |
| convert he using 1 | |
| rw [inv_pow,hsq,ENNReal.ofReal_inv_of_pos hcr,ENNReal.ofReal_coe_nnreal] | |
| apply measureFamilyModulus_lower_of_barycenter σ (fun v : (⋂ n, T n) => v.val) P _ hq hbar hc he' | |
| filter_upwards [hPs] with v hv | |
| exact ⟨⟨v.val,hv⟩,rfl⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |