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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
def allOpen (F : Set E) : Set (Set E) := {ω | F⊆ω}
lemma measurable_allOpen_block [Countable E] (F : Set E) :
MeasurableSet[blockSigma F] (allOpen F) := by
have he : allOpen F=⋂ e : F, {ω : Set E | (e:E)∈ω} := by
ext ω
simp only [allOpen,Set.mem_ofPred_eq,Set.mem_iInter]
exact ⟨fun h e => h e.property,fun h e he => h ⟨e,he⟩⟩
rw [he]
apply MeasurableSet.iInter
intro e
have hm : MeasurableSet[coordinateSigma (e:E)] {ω : Set E | (e:E)∈ω} :=
⟨{True},MeasurableSet.singleton _,by ext ω; simp⟩
exact (le_iSup_of_le (e:E) (le_iSup_of_le e.property le_rfl) :
coordinateSigma (e:E)≤blockSigma F) _ hm
lemma allOpen_inter_lower_le [Countable E] (q : E → unitInterval) (F : Set E)
(B : Set (Set E)) (hB : MeasurableSet B) (hBl : IsLowerSet B) :
(law q).real (allOpen F∩B)≤(law q).real (allOpen F)*(law q).real B := by
let C := forceOn F Set.univ ⁻¹' B
have hmC : MeasurableSet[blockSigma Fᶜ] C := hB.preimage (measurable_forceOn_block F _)
have hi := ((independent_blocks q F).indepSet_of_measurableSet
(measurable_allOpen_block F) hmC).measure_inter_eq_mul
have he : allOpen F∩B=allOpen F∩C := by
ext ω
by_cases hω : F⊆ω
· have hf : forceOn F Set.univ ω=ω := by
ext e
by_cases he : e∈F
· simp only [mem_forceOn,ite_eq_left he,Set.mem_univ,true_iff]
exact hω he
· simp only [mem_forceOn,ite_eq_right he]
simp only [Set.mem_inter_iff,allOpen,Set.mem_ofPred_eq,hω,true_and,C,Set.mem_preimage,hf]
· simp only [Set.mem_inter_iff,allOpen,Set.mem_ofPred_eq,hω,false_and]
have hsub : C⊆B := by
intro ω hω
apply hBl _ hω
intro e he
by_cases heF : e∈F <;> simp [heF,he]
have hr : (law q).real (allOpen F∩B)=(law q).real (allOpen F)*(law q).real C := by
simp only [measureReal_def,he,hi,ENNReal.toReal_mul]
rw [hr]
exact mul_le_mul_of_nonneg_left (measureReal_mono hsub) measureReal_nonneg
lemma integrable_blockProbability_real (q : E → unitInterval) (S : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) :
Integrable (fun σ => (blockProbability q S A σ).toReal) (law q) := by
have hm := (measurable_blockProbability q S A hA).mono (blockSigma_le S) le_rfl
apply (integrable_const (1:ℝ)).mono' hm.ennreal_toReal.aestronglyMeasurable
filter_upwards [] with σ
rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg]
exact ENNReal.toReal_mono (by simp) (blockProbability_le_one q S A σ)
lemma integral_blockProbability_real (q : E → unitInterval) (S : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) :
(∫ σ, (blockProbability q S A σ).toReal ∂law q)=(law q).real A := by
have hm := (measurable_blockProbability q S A hA).mono (blockSigma_le S) le_rfl
rw [integral_toReal hm.aemeasurable (ae_of_all _ (fun σ =>
(blockProbability_le_one q S A σ).trans_lt ENNReal.one_lt_top)),
lintegral_blockProbability q S A hA]
rfl
theorem good_trial_bound [Countable E] (q : E → unitInterval) (S F : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) (hAu : IsUpperSet A)
(η lam : ℝ) (hlam : 0<lam) (hmean : 1-η≤(law q).real A) :
(law q).real (allOpen F∩{σ | (blockProbability q S A σ).toReal<1-lam})≤
(law q).real (allOpen F)*η/lam := by
let g : Set E → ℝ := fun σ => (blockProbability q S A σ).toReal
let B : Set (Set E) := {σ | g σ<1-lam}
have hg : Measurable g :=
((measurable_blockProbability q S A hA).mono (blockSigma_le S) le_rfl).ennreal_toReal
have hgi : Integrable g (law q) := integrable_blockProbability_real q S A hA
have hg1 : ∀ σ, g σ≤1 := fun σ =>
ENNReal.toReal_mono (by simp) (blockProbability_le_one q S A σ)
have hgm : Monotone g := by
intro σ τ h
exact ENNReal.toReal_mono (ne_of_lt ((blockProbability_le_one q S A τ).trans_lt
ENNReal.one_lt_top)) (blockProbability_monotone q S A hAu h)
have hBm : MeasurableSet B := measurableSet_lt hg measurable_const
have hBl : IsLowerSet B := fun _ _ h hB => (hgm h).trans_lt hB
have hmark : lam*(law q).real B≤η := by
have hcomp : Integrable (fun σ => lam*B.indicator (1 : Set E → ℝ) σ) (law q) :=
((integrable_const (1:ℝ)).indicator hBm).const_mul lam
have hd : Integrable (fun σ => 1-g σ) (law q) := (integrable_const 1).sub hgi
have hi := integral_mono hcomp hd (fun σ => by
by_cases hσ : σ∈B
· rw [Set.indicator_of_mem hσ]
have hh : g σ<1-lam := hσ
dsimp only [Pi.one_apply]
linarith
· rw [Set.indicator_of_notMem hσ,mul_zero]
exact sub_nonneg.mpr (hg1 σ))
rw [integral_const_mul,integral_indicator_one hBm,integral_sub (integrable_const 1) hgi,
integral_const,probReal_univ,smul_eq_mul,one_mul] at hi
have he := integral_blockProbability_real q S A hA
change (∫ σ, g σ ∂law q)=(law q).real A at he
rw [he] at hi
linarith
have hh := allOpen_inter_lower_le q F B hBm hBl
have hb : (law q).real B≤η/lam := (le_div_iff₀ hlam).mpr (by nlinarith [hmark])
exact hh.trans (by simpa only [mul_div_assoc] using
mul_le_mul_of_nonneg_left hb (show 0≤(law q).real (allOpen F) from measureReal_nonneg))
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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