| import Mathlib.MeasureTheory.Measure.RegularityCompacts |
| import Mathlib.MeasureTheory.Constructions.Polish.Basic |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Set Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| / |
| theorem continuousOn_of_compact_graph {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| [T2Space X] {f : X → Y} {K : Set (X × Y)} (hK : IsCompact K) |
| (hgraph : ∀ z ∈ K, f z.1 = z.2) : ContinuousOn f (Prod.fst '' K) := by |
| let Q := Prod.fst '' K |
| let : CompactSpace K := isCompact_iff_compactSpace.mp hK |
| let g : K → X := fun z => z.1.1 |
| have hg : Continuous g := continuous_fst.comp continuous_subtype_val |
| have hi : Function.Injective g := by |
| intro z w h |
| apply Subtype.ext |
| apply Prod.ext h |
| exact (hgraph z.1 z.2).symm.trans ((congrArg f h).trans (hgraph w.1 w.2)) |
| have hmem (x : Q) : (x.1, f x.1) ∈ K := by |
| obtain ⟨z,hz,hx⟩ := x.2 |
| have he : (x.1, f x.1) = z := by |
| apply Prod.ext hx.symm |
| simpa only [← hx] using hgraph z hz |
| rwa [he] |
| let inv : Q → K := fun x => ⟨(x.1, f x.1), hmem x⟩ |
| have hinv : Continuous inv := |
| (hg.isClosedEmbedding hi).isEmbedding.isInducing.continuous_iff.mpr continuous_subtype_val |
| exact continuousOn_iff_continuous_domRestrict.mpr |
| (continuous_snd.comp (continuous_subtype_val.comp hinv)) |
|
|
| / |
| measurable graph. The compact set can be required to lie in a prescribed |
| Borel conull set. -/ |
| theorem exists_compact_lusin {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| [PolishSpace X] [PolishSpace Y] [MeasurableSpace X] [MeasurableSpace Y] |
| [BorelSpace X] [BorelSpace Y] (μ : Measure X) [IsFiniteMeasure μ] |
| {f : X → Y} (hf : Measurable f) {s : Set X} (hs : MeasurableSet s) |
| (hfull : ∀ᵐ x ∂μ, x ∈ s) {ε : ℝ≥0∞} (hε : ε ≠ 0) : |
| ∃ Q, Q ⊆ s ∧ IsCompact Q ∧ μ Qᶜ < ε ∧ ContinuousOn f Q := by |
| let graph : X → X × Y := fun x => (x,f x) |
| have hgraph : Measurable graph := measurable_id.prodMk hf |
| let G : Set (X × Y) := {z | z.1 ∈ s ∧ f z.1 = z.2} |
| have hG : MeasurableSet G := |
| (hs.preimage measurable_fst).inter (measurableSet_eq_fun (hf.comp measurable_fst) measurable_snd) |
| let ν := μ.map graph |
| obtain ⟨K,hKG,hK,hsmall⟩ := hG.exists_isCompact_sdiff_lt (μ := ν) (measure_ne_top _ _) hε |
| let Q : Set X := Prod.fst '' K |
| have hQ : IsCompact Q := hK.image continuous_fst |
| have hQs : Q ⊆ s := by |
| rintro x ⟨z,hz,rfl⟩ |
| exact (hKG hz).1 |
| have hmem (x : X) : graph x ∈ K ↔ x ∈ Q := by |
| constructor |
| · intro hx; exact ⟨graph x,hx,rfl⟩ |
| · rintro ⟨z,hz,hx⟩ |
| have he : graph x = z := by |
| apply Prod.ext hx.symm |
| simpa only [graph, ← hx] using (hKG hz).2 |
| rwa [he] |
| have hpre : graph ⁻¹' (G \ K) = s \ Q := by |
| ext x |
| change ((x ∈ s ∧ f x = f x) ∧ graph x ∉ K) ↔ (x ∈ s ∧ x ∉ Q) |
| rw [hmem] |
| simp |
| have hsmall' : μ (s \ Q) < ε := by |
| have he : ν (G \ K) = μ (s \ Q) := by |
| rw [Measure.map_apply hgraph (hG.diff hK.isClosed.measurableSet), hpre] |
| rwa [he] at hsmall |
| have heq : μ (s \ Q) = μ Qᶜ := by |
| apply measure_congr |
| filter_upwards [hfull] with x hx |
| apply propext |
| change (x ∈ s ∧ x ∉ Q) ↔ x ∉ Q |
| simp [hx] |
| refine ⟨Q,hQs,hQ,heq ▸ hsmall',?_⟩ |
| exact continuousOn_of_compact_graph hK (fun z hz => (hKG hz).2) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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|
