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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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