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f205175 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory
open scoped Classical
namespace CriticalPercolation
variable {Ω I : Type*} [MeasurableSpace Ω]
def firstTest (P : ℕ → Ω → Prop) (ω : Ω) : ℕ :=
if h : ∃ n, P n ω then Nat.find h else 0
omit [MeasurableSpace Ω] in
lemma firstTest_eq (P : ℕ → Ω → Prop) (ω : Ω) (n : ℕ) :
firstTest P ω=n ↔ (P n ω ∧ ∀ m<n, ¬P m ω) ∨ ((¬∃ m, P m ω) ∧ n=0) := by
by_cases h : ∃ m, P m ω
· simp [firstTest,h,Nat.find_eq_iff]
· have hn : ¬P n ω := fun hn => h ⟨n,hn⟩
simp [firstTest,h,hn,eq_comm]
lemma measurableSet_firstTest (P : ℕ → Ω → Prop) (hP : ∀ n, MeasurableSet {ω | P n ω}) (n : ℕ) :
MeasurableSet {ω | firstTest P ω=n} := by
simp_rw [firstTest_eq]
apply MeasurableSet.union
· exact (hP n).inter ((Measurable.forall (fun m => measurable_const.imp (hP m).mem.not)).setOf)
· exact (Measurable.exists (fun m => (hP m).mem)).setOf.compl.inter (MeasurableSet.const _)
theorem exists_honest_selector [Countable I] [Nonempty I] (P : Ω → I → Prop)
(hP : ∀ i, MeasurableSet {ω | P ω i}) :
∃ d : Ω → I, (∀ i, MeasurableSet {ω | d ω=i}) ∧
∀ ω, (∃ i, P ω i) → P ω (d ω) := by
obtain ⟨e,he⟩ := exists_surjective_nat I
let d : Ω → I := fun ω => e (firstTest (fun n ω => P ω (e n)) ω)
refine ⟨d,?_,?_⟩
· intro i
have hh : {ω | d ω=i}=⋃ n : ℕ, ({ω | firstTest (fun n ω => P ω (e n)) ω=n}∩{_ω | e n=i}) := by
ext ω
simp only [Set.mem_iUnion,Set.mem_inter_iff,Set.mem_ofPred_eq]
exact ⟨fun h => ⟨_,rfl,h⟩,fun ⟨n,hn,hni⟩ => by dsimp [d]; rw [hn,hni]⟩
rw [hh]
exact MeasurableSet.iUnion (fun n => (measurableSet_firstTest _ (fun m => hP (e m)) n).inter
(MeasurableSet.const _))
· intro ω ⟨i,hi⟩
obtain ⟨n,rfl⟩ := he i
have hex : ∃ n, P ω (e n) := ⟨n,hi⟩
dsimp [d,firstTest]
rw [dite_eq_left hex]
exact Nat.find_spec hex
end CriticalPercolation
end
end
end
end OAI
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