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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness
namespace OAI
section
section
open scoped BigOperators
open scoped Classical ENNReal
open Set
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Filter Topology
open scoped Topology Classical
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma measurableSet_restrict (s : Set E) (hs : s.Finite) (A : Set (Set E)) :
MeasurableSet {ω | s ∩ ω ∈ A} := by
let := hs.fintype
have hf : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
Measurable.of_eval (fun edge => measurable_set_mem edge.val)
have he : {ω : Set E | s ∩ ω ∈ A} =
(fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) ⁻¹'
{u : Set s | Subtype.val '' u ∈ A} := by
ext ω
have hi : Subtype.val '' (Subtype.val ⁻¹' ω : Set s) = s ∩ ω := by
ext e
simp
simp only [Set.mem_ofPred_eq, Set.mem_preimage, hi]
rw [he]
exact hf (.of_discrete)
lemma measurableSet_finiteWitness [Countable E] (A : Set (Set E))
(hmono : IsUpperSet A) (hw : HasFiniteOpenWitness A) : MeasurableSet A := by
let K := (Set.to_countable (Set.univ : Set E)).finiteExhaustion
rw [← finiteWitness_exhaustion A hmono hw K]
exact MeasurableSet.iUnion fun n => measurableSet_restrict (K n) (K.finite n) A
end CriticalPercolation.BernoulliProduct
end
end
end OAI
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