| 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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|
