File size: 3,553 Bytes
ddc84c5 | 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 | import Mathlib.MeasureTheory.Measure.RegularityCompacts
import Mathlib.MeasureTheory.Constructions.Polish.Basic
namespace OAI
namespace FiniteFactor
open MeasureTheory Set Filter
open scoped ENNReal
noncomputable section
/-- A function whose graph is compact above Q is continuous on Q. -/
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))
/-- Lusin's theorem on Polish spaces, obtained from compact subsets of the
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
|