| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BernoulliProduct |
| variable {E I : Type*} |
|
|
| lemma block_event_disjoint_forceOn (X Y : Set E) (hXY : Disjoint X Y) |
| (A : Set (Set E)) (hA : MeasurableSet[blockSigma Y] A) (σ ω : Set E) : |
| forceOn X σ ω∈A ↔ ω∈A := by |
| have he : forceOn Y (forceOn X σ ω) ω=ω := by |
| ext e |
| by_cases heY : e∈Y |
| · have heX : e∉X := fun h => Set.disjoint_left.mp hXY h heY |
| simp only [mem_forceOn,ite_eq_left heY,ite_eq_right heX] |
| · simp only [mem_forceOn,ite_eq_right heY] |
| have hh := block_event_forceOn Y A hA (forceOn X σ ω) ω |
| rw [he] at hh |
| exact hh.symm |
|
|
| def selectedEvent (d : Set E → I) (A : I → Set (Set E)) : Set (Set E) := |
| {ω | ω∈A (d ω)} |
|
|
| lemma selectedEvent_eq_iUnion (d : Set E → I) (A : I → Set (Set E)) : |
| selectedEvent d A=⋃ i, ({ω | d ω=i}∩A i) := by |
| ext ω |
| simp only [selectedEvent,Set.mem_ofPred_eq,Set.mem_iUnion,Set.mem_inter_iff] |
| exact ⟨fun h => ⟨d ω,rfl,h⟩,fun ⟨i,hi,h⟩ => hi.symm ▸ h⟩ |
|
|
| lemma measurableSet_selectedEvent [Countable I] (d : Set E → I) |
| (A : I → Set (Set E)) (hd : ∀ i, MeasurableSet {ω | d ω=i}) |
| (hA : ∀ i, MeasurableSet (A i)) : MeasurableSet (selectedEvent d A) := by |
| rw [selectedEvent_eq_iUnion] |
| exact MeasurableSet.iUnion (fun i => (hd i).inter (hA i)) |
|
|
| theorem blockProbability_selected (q : E → unitInterval) (X Y : Set E) |
| (hXY : Disjoint X Y) (d : Set E → I) (A : I → Set (Set E)) |
| (hd : ∀ i, MeasurableSet[blockSigma X] {ω | d ω=i}) |
| (hA : ∀ i, MeasurableSet[blockSigma Y] (A i)) (σ : Set E) : |
| blockProbability q X (selectedEvent d A) σ=law q (A (d σ)) := by |
| unfold blockProbability |
| congr 1 |
| ext ω |
| change forceOn X σ ω∈A (d (forceOn X σ ω)) ↔ ω∈A (d σ) |
| have hh : d (forceOn X σ ω)=d σ := |
| (block_event_forceOn X {τ | d τ=d σ} (hd _) σ ω).mpr rfl |
| rw [hh] |
| exact block_event_disjoint_forceOn X Y hXY (A (d σ)) (hA _) σ ω |
|
|
| theorem selected_fresh_bound [Countable I] (q : E → unitInterval) (X Y : Set E) |
| (hXY : Disjoint X Y) (d : Set E → I) (A : I → Set (Set E)) |
| (hd : ∀ i, MeasurableSet[blockSigma X] {ω | d ω=i}) |
| (hA : ∀ i, MeasurableSet[blockSigma Y] (A i)) |
| (C : Set (Set E)) (hC : MeasurableSet[blockSigma X] C) |
| (r : ℝ) (hr : 0 ≤ r) (hb : ∀ σ∈C, (law q).real (A (d σ)) ≤ r) : |
| (law q).real (C∩selectedEvent d A) ≤ r*(law q).real C := by |
| have hmA := measurableSet_selectedEvent d A (fun i => blockSigma_le X _ (hd i)) |
| (fun i => blockSigma_le Y _ (hA i)) |
| have hk (σ : Set E) (hσ : σ∈C) : |
| blockProbability q X (selectedEvent d A) σ ≤ ENNReal.ofReal r := by |
| rw [blockProbability_selected q X Y hXY d A hd hA σ] |
| apply (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mp |
| rw [ENNReal.toReal_ofReal hr] |
| exact hb σ hσ |
| have hh : law q (selectedEvent d A∩C) ≤ ENNReal.ofReal r*law q C := by |
| rw [← setLIntegral_blockProbability q X _ C hmA hC] |
| calc |
| (∫⁻ σ in C, blockProbability q X (selectedEvent d A) σ ∂law q) ≤ |
| ∫⁻ _ in C, ENNReal.ofReal r ∂law q := |
| setLIntegral_mono' (blockSigma_le X _ hC) hk |
| _ = _ := by rw [lintegral_const,Measure.restrict_apply_univ] |
| have ht := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top |
| (measure_ne_top _ _)) hh |
| simpa only [Set.inter_comm (selectedEvent d A) C,ENNReal.toReal_mul, |
| ENNReal.toReal_ofReal hr,Measure.real] using ht |
|
|
| end CriticalPercolation.BernoulliProduct |
| end |
| end |
| end |
|
|
| end OAI |
|
|
