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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 61 62 63 64 65 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.ClosedCutProbability
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
private lemma forceOn_mem_iff (S σ ω : Set E) (edge : E) :
edge ∈ forceOn S σ ω ↔ if edge ∈ S then edge ∈ σ else edge ∈ ω := Iff.rfl
lemma forceOn_insert (S : Set E) (e : E) (σ ω : Set E) :
forceOn (insert e S) σ ω = forceOn {e} σ (forceOn S σ ω) := by
classical
ext edge
simp only [forceOn_mem_iff]
by_cases hedge : edge = e <;> simp [hedge]
lemma forceOn_empty_eq_sdiff (S ω : Set E) : forceOn S ∅ ω = ω \ S := by
classical
ext e
show (if e∈S then e∈(∅ : Set E) else e∈ω) ↔ _
split_ifs <;> simp_all
lemma finite_forceOn_preimage_null (q : E → unitInterval)
(hq : ∀ e, (q e : ℝ)∈Set.Ioo 0 1) (S : Set E) (hS : S.Finite) (σ : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) (hnull : law q A=0) :
law q (forceOn S σ ⁻¹' A)=0 := by
classical
induction S,hS using Set.Finite.induction_on generalizing A with
| empty =>
have he : forceOn (∅ : Set E) σ = id := by
funext ω
ext edge
simp only [forceOn_mem_iff, Set.mem_empty_iff_false, ite_false, id_eq]
rwa [he,Set.preimage_id_eq]
| @insert e S he hS ih =>
have hb : bitMeasure (q e) {e∈σ} ≠ 0 := by
by_cases h : e∈σ
· simpa only [h] using bitMeasure_true_ne_zero (q e) (hq e).1
· simpa only [h] using bitMeasure_false_ne_zero (q e) (hq e).2
have hi := force_singleton_preimage_null q e σ hb A hA hnull
have hm : Measurable (forceOn ({e} : Set E) σ) :=
(measurable_forceOn_block {e} σ).mono (blockSigma_le _) le_rfl
have hh := ih (forceOn {e} σ ⁻¹' A) (hA.preimage hm) hi
simpa only [Set.preimage_preimage,Function.comp_def,←forceOn_insert] using hh
lemma ae_finite_forceOn (q : E → unitInterval)
(hq : ∀ e, (q e : ℝ)∈Set.Ioo 0 1) (S : Set E) (hS : S.Finite) (σ : Set E)
(P : Set E → Prop) (hm : Measurable P) (h : ∀ᵐ ω ∂law q, P ω) :
∀ᵐ ω ∂law q, P (forceOn S σ ω) := by
rw [ae_iff] at h ⊢
exact finite_forceOn_preimage_null q hq S hS σ {ω | ¬P ω} hm.not.setOf h
end CriticalPercolation.BernoulliProduct
end
end
end OAI
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