| 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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|
