openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /BorelImageCompactModulus.lean
| import OAI.Analysis.CircleDomains.Modulus.CompactContinuousImageModulus | |
| import OAI.Analysis.CircleDomains.Probability.PositiveMassCompactLocalization | |
| import Mathlib.MeasureTheory.Constructions.Polish.Basic | |
| namespace OAI | |
| /-! Compact inner approximation for the pointwise modulus of a Borel image. | |
| The compatible Polish topology is refined only after compact localization. -/ | |
| open Set Function Filter MeasureTheory TopologicalSpace | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_compact_subfamily_of_borel_image | |
| {A Z : Type*} [MeasurableSpace A] [StandardBorelSpace A] | |
| [TopologicalSpace Z] [PolishSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| (σ : Measure Z) [IsFiniteMeasure σ] (K : A → FiniteMeasure Z) (hKm : Measurable K) | |
| (hK : ∀ a, K a ≠ 0) {c : ℝ≥0∞} | |
| (hc : c < measureFamilyModulus A Z σ K) : | |
| ∃ S : Set (FiniteMeasure Z), IsCompact S ∧ S ⊆ range K ∧ | |
| c < measureFamilyModulus S Z σ (fun v => v.val) := by | |
| classical | |
| obtain ⟨T,hT,hTz,hmod⟩ := exists_compact_nonzero_modulus_localization σ K hK hc | |
| let B : Set A := {a | K a ∈ T} | |
| have hBm : MeasurableSet B := hKm (compact_finiteMeasure_measurableSet hT) | |
| let : StandardBorelSpace B := hBm.standardBorel | |
| let := upgradeStandardBorel B | |
| let : BorelSpace T := compact_finiteMeasure_borel hT | |
| let : PolishSpace T := compact_finiteMeasure_polish hT | |
| let f : B → T := fun a => ⟨K a.val,a.property⟩ | |
| have hfm : Measurable f := (hKm.comp measurable_subtype_coe).subtype_mk | |
| obtain ⟨t,hle,hfc,ht⟩ := hfm.exists_continuous | |
| have hbe : @BorelSpace B t inferInstance := by | |
| constructor | |
| rw [eq_borel_upgradeStandardBorel B,borel_eq_borel_of_le ht _ hle] | |
| infer_instance | |
| let : TopologicalSpace B := t | |
| let : PolishSpace B := ht | |
| let : BorelSpace B := hbe | |
| let := upgradeIsCompletelyMetrizable B | |
| cases isEmpty_or_nonempty B with | |
| | inl hB => | |
| have hz := measureFamilyModulus_zero_of_isEmpty σ (fun a : B => K a.val) | |
| rw [hz] at hmod | |
| exact (not_lt_of_ge bot_le hmod).elim | |
| | inr hB => | |
| obtain ⟨S,hS,hSf,hSc⟩ := exists_compact_subfamily_of_continuous_image σ hT hTz f hfc hmod | |
| refine ⟨S,hS,?_,hSc⟩ | |
| rintro v hv | |
| obtain ⟨a,ha⟩ := hSf hv | |
| exact ⟨a.val,ha⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |