File size: 2,751 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.BernoulliSingleton
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma integral_singleton_decomposition (q : E → unitInterval) (e : E)
(f : Set E → ℝ) (hm : Measurable f) (C : ℝ) (hb : ∀ ω, ‖f ω‖≤C) :
(∫ ω, f ω ∂law q) = (q e : ℝ)*(∫ ω, f (forceOn {e} Set.univ ω) ∂law q)+
(1-(q e : ℝ))*(∫ ω, f (forceOn {e} ∅ ω) ∂law q) := by
have hi (σ : Set E) : Integrable f ((law q).map (forceOn {e} σ)) := by
exact Integrable.of_bound hm.aestronglyMeasurable C (Eventually.of_forall hb)
conv_lhs => rw [law_singleton_decomposition q e]
rw [integral_add_measure ((hi _).smul_measure (measure_ne_top _ _))
((hi _).smul_measure (measure_ne_top _ _)),integral_smul_measure,integral_smul_measure,
integral_map (measurable_forceOn _ _).aemeasurable hm.aestronglyMeasurable,
integral_map (measurable_forceOn _ _).aemeasurable hm.aestronglyMeasurable]
have ht := bitMeasure_real_singleton (q e) True
have hf := bitMeasure_real_singleton (q e) False
simp only [measureReal_def,bitWeight,ite_true,ite_false] at ht hf
simp only [smul_eq_mul,ht,hf]
lemma integral_centered_singleton (q : E → unitInterval) (e : E)
(f : Set E → ℝ) (hm : Measurable f) (C : ℝ) (hC : 0≤C) (hb : ∀ ω, ‖f ω‖≤C) :
(∫ ω, (if e∈ω then 1-(q e : ℝ) else -(q e : ℝ))*f ω ∂law q) =
(q e : ℝ)*(1-(q e : ℝ))*
((∫ ω, f (forceOn {e} Set.univ ω) ∂law q) - (∫ ω, f (forceOn {e} ∅ ω) ∂law q)) := by
have hm' : Measurable (fun ω => (if e∈ω then 1-(q e : ℝ) else -(q e : ℝ))*f ω) :=
(measurable_const.ite (measurable_set_mem e).setOf measurable_const).mul hm
have hb' (ω : Set E) : ‖(if e∈ω then 1-(q e : ℝ) else -(q e : ℝ))*f ω‖≤C := by
rw [norm_mul]
have hc : ‖(if e∈ω then 1-(q e : ℝ) else -(q e : ℝ))‖≤1 := by
split_ifs <;> simp only [Real.norm_eq_abs]
· rw [abs_of_nonneg (sub_nonneg.mpr (q e).property.2)]
linarith [(q e).property.1]
· rw [abs_neg,abs_of_nonneg (q e).property.1]
exact (q e).property.2
exact (mul_le_mul_of_nonneg_left (hb ω) (norm_nonneg _)).trans
((mul_le_mul_of_nonneg_right hc hC).trans_eq (one_mul C))
rw [integral_singleton_decomposition q e _ hm' C hb']
have ht (ω : Set E) : e∈forceOn {e} Set.univ ω := by simp
have hf (ω : Set E) : e∉forceOn {e} ∅ ω := by simp
simp only [ite_eq_left (ht _),ite_eq_right (hf _),integral_const_mul]
ring
end CriticalPercolation.BernoulliProduct
end
end
end OAI
|