| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E B : Type*} | |
| lemma measurable_block_of_dependsOn {A : Type*} [MeasurableSpace A] | |
| {f : Set E → A} (hf : Measurable f) (S : Set E) (hdep : DependsOn f S) : | |
| @Measurable (Set E) A (blockSigma S) inferInstance f := by | |
| have hm := hf.comp (measurable_forceOn_block Sᶜ (∅ : Set E)) | |
| rw [compl_compl] at hm | |
| convert hm using 1 | |
| funext ω | |
| apply hdep | |
| intro e he | |
| simp only [forceOn, Set.mem_compl_iff, ite_eq_right (not_not.mpr he), | |
| Set.mem_empty_iff_false] | |
| rfl | |
| lemma lintegral_indicator_fresh (q : E → unitInterval) (S : Set E) | |
| (A : Set (Set E)) (hA : MeasurableSet A) (hdep : DependsOn (fun ω => ω ∈ A) S) | |
| (g : Set E → ℝ≥0∞) (hg : Measurable g) (hgdep : DependsOn g Sᶜ) : | |
| (∫⁻ ω, A.indicator g ω ∂law q) = law q A * ∫⁻ ω, g ω ∂law q := by | |
| have hAb : MeasurableSet[blockSigma S] A := by | |
| have hm := hA.preimage (measurable_forceOn_block Sᶜ (∅ : Set E)) | |
| rw [compl_compl] at hm | |
| convert hm using 1 | |
| ext ω | |
| apply Iff.of_eq | |
| apply hdep | |
| intro e he | |
| simp only [forceOn, Set.mem_compl_iff, ite_eq_right (not_not.mpr he)] | |
| rfl | |
| have hgb := measurable_block_of_dependsOn hg Sᶜ hgdep | |
| have hi := lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace | |
| (blockSigma_le S) (blockSigma_le Sᶜ) | |
| (independent_blocks q S) | |
| ((measurable_const : @Measurable (Set E) ℝ≥0∞ (blockSigma S) inferInstance | |
| (fun _ => 1)).indicator hAb) hgb | |
| have heq : (fun ω => A.indicator (fun _ => (1 : ℝ≥0∞)) ω * g ω) = A.indicator g := by | |
| funext ω | |
| by_cases hω : ω ∈ A <;> simp [Set.indicator, hω] | |
| have hone := lintegral_indicator_const (μ := law q) hA (1 : ℝ≥0∞) | |
| simp only [one_mul] at hone | |
| rw [heq, hone] at hi | |
| exact hi | |
| theorem lintegral_countable_exposure [Countable B] [MeasurableSpace B] | |
| [MeasurableSingletonClass B] (q : E → unitInterval) (exposed : Set E → B) | |
| (hm : Measurable exposed) (block : B → Set E) | |
| (hd : ∀ b, DependsOn (fun ω => exposed ω = b) (block b)) | |
| (g : B → Set E → ℝ≥0∞) (hg : ∀ b, Measurable (g b)) | |
| (hgd : ∀ b, DependsOn (g b) (block b)ᶜ) : | |
| (∫⁻ ω, g (exposed ω) ω ∂law q) = | |
| ∑' b, law q {ω | exposed ω = b} * ∫⁻ η, g b η ∂law q := by | |
| have hA (b : B) : MeasurableSet {ω | exposed ω = b} := | |
| hm (MeasurableSet.singleton b) | |
| have heq (ω : Set E) : g (exposed ω) ω = | |
| ∑' b, ({ω | exposed ω = b} : Set (Set E)).indicator (g b) ω := by | |
| simp [Set.indicator, eq_comm] | |
| simp_rw [heq] | |
| rw [lintegral_tsum (fun b => ((hg b).indicator (hA b)).aemeasurable)] | |
| apply tsum_congr | |
| intro b | |
| exact lintegral_indicator_fresh q (block b) _ (hA b) (hd b) (g b) (hg b) (hgd b) | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end OAI | |