lean/OAI/Probability/CriticalPercolation/ProductMeasure/MeasurableSetRestrict.lean

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