openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CompactFiniteMeasureBorel.lean
| import OAI.Analysis.CircleDomains.Probability.CompactFiniteMeasureMeasurability | |
| import OAI.Analysis.CircleDomains.Probability.CompactProbabilityBorel | |
| import OAI.Analysis.CircleDomains.Selection.CompleteEmbeddingGDelta | |
| namespace OAI | |
| /-! Compact families of finite measures retain their original evaluation | |
| sigma algebra. A countable continuous and measurable code proves both | |
| compact-set measurability and the Borel structure on each compact family. -/ | |
| open Set Function Filter MeasureTheory TopologicalSpace Topology | |
| open scoped Topology NNReal BoundedContinuousFunction | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_finiteMeasure_countable_code | |
| (Z : Type*) [TopologicalSpace Z] [PolishSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] : | |
| ∃ e : FiniteMeasure Z → (ℕ → ℝ≥0), Continuous e ∧ Measurable e ∧ Injective e := by | |
| let : MetricSpace Z := metrizableSpaceMetric Z | |
| obtain ⟨f,hf⟩ := Metric.PiNatEmbed.exists_embedding_to_hilbert_cube (X := Z) | |
| have hfm : MeasurableEmbedding f := hf.continuous.measurableEmbedding hf.injective | |
| let X := ℕ → unitInterval | |
| let : SeparableSpace (X →ᵇ ℝ≥0) := | |
| (ContinuousMap.isometryEquivBoundedOfCompact X ℝ≥0).surjective.denseRange.separableSpace | |
| (ContinuousMap.isometryEquivBoundedOfCompact X ℝ≥0).continuous | |
| obtain ⟨b,hb⟩ := exists_dense_seq (X →ᵇ ℝ≥0) | |
| let e : FiniteMeasure Z → (ℕ → ℝ≥0) := | |
| fun v n => (v.map f).testAgainstNN (b n) | |
| have hec : Continuous e := continuous_pi (fun n => | |
| (FiniteMeasure.continuous_testAgainstNN_eval (b n)).comp | |
| (FiniteMeasure.continuous_map hf.continuous)) | |
| have hem : Measurable e := by | |
| apply Measurable.of_eval | |
| intro n | |
| have hm : Measurable (fun v : FiniteMeasure Z => | |
| (v.map f : Measure X)) := | |
| (Measure.measurable_map f hfm.measurable).comp measurable_subtype_coe | |
| exact (Measure.measurable_lintegral | |
| ((ENNReal.continuous_coe.comp (b n).continuous).measurable)).ennreal_toNNReal.comp hm | |
| refine ⟨e,hec,hem,?_⟩ | |
| intro v w heq | |
| have hm : v.map f = w.map f := by | |
| apply FiniteMeasure.injective_toWeakDualBCNN | |
| apply DFunLike.ext | |
| intro g | |
| exact hb.induction_on g | |
| (isClosed_eq (v.map f).testAgainstNN_lipschitz.continuous | |
| (w.map f).testAgainstNN_lipschitz.continuous) | |
| (fun n => congrFun heq n) | |
| apply FiniteMeasure.toMeasure_injective | |
| exact hfm.map_injective (congrArg FiniteMeasure.toMeasure hm) | |
| lemma compact_finiteMeasure_measurableSet | |
| {Z : Type*} [TopologicalSpace Z] [PolishSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] {S : Set (FiniteMeasure Z)} | |
| (hS : IsCompact S) : MeasurableSet S := by | |
| obtain ⟨e,hec,hem,hei⟩ := exists_finiteMeasure_countable_code Z | |
| have hclosed : IsClosed (e '' S) := (hS.image hec).isClosed | |
| convert hem hclosed.measurableSet using 1 | |
| exact (preimage_image_eq S hei).symm | |
| lemma compact_finiteMeasure_borel | |
| {Z : Type*} [TopologicalSpace Z] [PolishSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] {S : Set (FiniteMeasure Z)} | |
| (hS : IsCompact S) : BorelSpace S := by | |
| let : CompactSpace S := isCompact_iff_compactSpace.mp hS | |
| obtain ⟨e,hec,hem,hei⟩ := exists_finiteMeasure_countable_code Z | |
| have he : IsClosedEmbedding (fun v : S => e v.val) := | |
| (hec.comp continuous_subtype_val).isClosedEmbedding (hei.comp Subtype.val_injective) | |
| have hm : Measurable (fun v : S => e v.val) := hem.comp measurable_subtype_coe | |
| refine ⟨le_antisymm (compact_finiteMeasure_measurable_le_borel S) ?_⟩ | |
| rw [he.isEmbedding.eq_induced,borel_comap,← BorelSpace.measurable_eq] | |
| exact hm.comap_le | |
| lemma compact_finiteMeasure_polish | |
| {Z : Type*} [TopologicalSpace Z] [PolishSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] {S : Set (FiniteMeasure Z)} | |
| (hS : IsCompact S) : PolishSpace S := by | |
| let : CompactSpace S := isCompact_iff_compactSpace.mp hS | |
| obtain ⟨e,hec,_,hei⟩ := exists_finiteMeasure_countable_code Z | |
| have he : IsClosedEmbedding (fun v : S => e v.val) := | |
| (hec.comp continuous_subtype_val).isClosedEmbedding (hei.comp Subtype.val_injective) | |
| exact he.polishSpace | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |