| import Mathlib |
|
|
| namespace OAI |
| /-! Borel selection for a closed relation into a compact metric space. -/ |
| noncomputable section |
| open Set Filter Metric Real MeasureTheory |
| open scoped Topology NNReal ENNReal |
| namespace GeneralMahler.Select |
| section |
| variable {X Y:Type*} |
| |
| variable (F:Set (X×Y)) |
| def near (x:X) (y:Y) (r:ℝ) := ∃ v,(x,v)∈F∧dist v y≤r |
| omit in |
| lemma near_m (h:IsClosed F) (y:Y) (r:ℝ) : MeasurableSet {x|near F x y r} := by |
| let S := F ∩ Prod.snd ⁻¹' (closedBall y r) |
| have hs := isClosedMap_fst_of_compactSpace S (h.inter (isClosed_closedBall.preimage continuous_snd)) |
| have he : {x|near F x y r}= Prod.fst '' S := by |
| ext x |
| constructor |
| · rintro ⟨v,hv,h⟩; exact ⟨(x,v),⟨hv,h⟩,rfl⟩ |
| rintro ⟨⟨x,v⟩,h,rfl⟩; exact ⟨v,h.1,h.2⟩ |
| rw ; exact hs.measurableSet |
| def eps (n:ℕ):ℝ := (1/2)^n |
| omit in |
| lemma dense_approx (y:Y) {r:ℝ} (hr:0<r) : |
| ∃ n:ℕ,dist y (TopologicalSpace.denseSeq Y n)<r := |
| (TopologicalSpace.denseRange_denseSeq Y).exists_dist_lt y hr |
| omit in |
| lemma choose_m {p:X→Y→Prop} (h:∀ x,∃ n:ℕ,p x (TopologicalSpace.denseSeq Y n)) |
| (hm:∀ n:ℕ,MeasurableSet {x|p x (TopologicalSpace.denseSeq Y n)}) : |
| ∃ f:X→Y,Measurable f∧∀ x,p x (f x) := by |
| classical |
| let f := fun x:X=> Nat.find (h x) |
| exact ⟨TopologicalSpace.denseSeq Y ∘ f,(measurable_of_countable _).comp |
| (measurable_find h hm),fun x=> Nat.find_spec (h x)⟩ |
|
|
| abbrev tag (n:ℕ) := {f:X→Y // Measurable f∧∀ x, near F x (f x) (eps n)} |
| omit in |
| lemma start (he:∀ x:X,∃ y:Y,(x,y)∈F) (hc:IsClosed F) : |
| Nonempty (tag F 0) := by |
| have h (x:X) : ∃ n:ℕ,near F x (TopologicalSpace.denseSeq Y n) (eps 0) := by |
| obtain ⟨y,hy⟩ := he x |
| obtain ⟨n,hn⟩ := dense_approx y (r:=eps 0) (by norm_num ) |
| exact ⟨n,y,hy,hn.le⟩ |
| obtain ⟨f,hf,hf'⟩ := choose_m (p:=fun x y=> near F x y (eps 0)) h (fun _=> near_m _ hc _ _) |
| exact ⟨⟨f,hf,hf'⟩⟩ |
| lemma next (hc:IsClosed F) (n:ℕ) (x:tag F n) : |
| ∃ y: tag F (n+1),∀ a,dist (x.val a) (y.val a)≤2*eps n := by |
| let p := fun (a:X) (y:Y)=> near F a y (eps (n+1)) ∧ dist (x.val a) y≤2*eps n |
| have h (a:X) : ∃ n,p a (TopologicalSpace.denseSeq Y n) := by |
| obtain ⟨v,hv,hh⟩ := x.property.2 a |
| obtain ⟨w,hw⟩ := dense_approx v (r:=eps (n+1)) (by unfold eps; positivity) |
| refine ⟨w,⟨v,hv,hw.le⟩,?_⟩ |
| apply le_trans (dist_triangle _ v _) |
| have hp : 0≤eps n := by unfold eps; positivity |
| have he : eps (n+1)=eps n/2 := by unfold eps; rw ; ring |
| rw ; linarith |
| have hm (i) : MeasurableSet {a|p a (TopologicalSpace.denseSeq Y i)} := |
| (near_m _ hc _ _).inter (measurableSet_le (x.property.1.dist measurable_const) measurable_const) |
| obtain ⟨f,hf,h₁⟩ := choose_m h hm |
| exact ⟨⟨f,hf,fun x=>(h₁ x).1⟩,fun x=>(h₁ x).2⟩ |
|
|
| theorem selector (he:∀ x:X,∃ y:Y,(x,y)∈F) (hc:IsClosed F) : |
| ∃ f:X→Y, Measurable f ∧ ∀ x,(x,f x)∈F := by |
| classical |
| let s (n:ℕ) := Nat.rec (motive:=fun n=> tag F n) (Classical.choice (start F he hc)) |
| (fun n x=> (next F hc n x).choose) n |
| have hn (n a) : dist ((s n).val a) ((s (n+1)).val a) ≤ 2*eps n := (next F hc n (s n)).choose_spec a |
| let g := fun n=>(s n).val |
| have hh (x:X) : CauchySeq fun n=>g n x := |
| cauchySeq_of_le_geometric _ _ (show (1/2:ℝ)<1 by norm_num) (fun n=>hn n x) |
| choose f hf using fun x:X=> cauchySeq_tendsto_of_complete (hh x) |
| refine ⟨f,measurable_of_tendsto_metrizable (fun n=>(s n).property.1) (tendsto_pi_nhds.mpr hf),?_⟩ |
| intro x |
| choose y hy h using fun n=> (s n).property.2 x |
| have he : Tendsto y atTop (𝓝 (f x)) := by |
| rw |
| have h₁ := tendsto_pow_atTop_nhds_zero_of_lt_one (show (0:ℝ) ≤ 1/2 by norm_num) (by norm_num) |
| have h₂ := tendsto_iff_dist_tendsto_zero.mp (hf x) |
| refine squeeze_zero (g:=fun n=>eps n+dist (g n x) (f x)) (fun _=>dist_nonneg) (fun n=> ?_) ?_ |
| · exact le_trans (dist_triangle (y n) (g n x) (f x)) (add_le_add (h n) le_rfl) |
| simpa using h₁.add h₂ |
| exact hc.mem_of_tendsto (tendsto_const_nhds.prodMk_nhds he) (Eventually.of_forall hy) |
| end |
| end GeneralMahler.Select |
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| end |
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| end OAI |
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