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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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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