File size: 3,785 Bytes
67aee1d | 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 | import OAI.Analysis.FiniteFactor.Sampling.Rectangle
import Mathlib.MeasureTheory.Integral.Layercake
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal
noncomputable section
/-- A bounded measurable real function on a finite measure space is integrable. -/
theorem sampling_integrable_bounded {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (f : Ω → ℝ) (hf : Measurable f)
(C : ℝ) (hb : ∀ x, |f x| ≤ C) : Integrable f μ := by
apply (integrable_const C).mono' hf.aestronglyMeasurable
exact Eventually.of_forall (fun x => by simpa [Real.norm_eq_abs] using hb x)
/-- Finite layer-cake formula for a product, valid even for a singular joint law. -/
theorem sampling_product_layerCake {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (f g : Ω → ℝ)
(hf : Measurable f) (hg : Measurable g) (C : ℝ) (_hC : 0 ≤ C)
(hfb : ∀ x, 0 ≤ f x ∧ f x ≤ C) (hgb : ∀ x, 0 ≤ g x ∧ g x ≤ C) :
(∫ x, f x*g x ∂μ) =
∫ r in Set.Ioc 0 C, ∫ t in Set.Ioc 0 C,
(μ {x | r ≤ f x ∧ t ≤ g x}).toReal := by
let ν := volume.restrict (Set.Ioc 0 C)
let K : (ℝ × ℝ) × Ω → ℝ := fun p =>
if p.1.1 ≤ f p.2 ∧ p.1.2 ≤ g p.2 then 1 else 0
have hK : Measurable K := measurable_const.ite
((measurableSet_le measurable_fst.fst (hf.comp measurable_snd)).inter
(measurableSet_le measurable_fst.snd (hg.comp measurable_snd))) measurable_const
have hKb (p) : |K p| ≤ 1 := by dsimp [K]; split_ifs <;> norm_num
have hKi := sampling_integrable_bounded ((ν.prod ν).prod μ) K hK 1 hKb
have hpoint (x : Ω) : (∫ p, K (p,x) ∂ν.prod ν) = f x*g x := by
have hik : Integrable (fun p => K (p,x)) (ν.prod ν) :=
sampling_integrable_bounded (ν.prod ν) _
(hK.comp (measurable_id.prodMk measurable_const)) 1 (fun p => hKb (p,x))
rw [integral_prod _ hik]
have he (r t : ℝ) : K ((r,t),x) =
(Set.Iic (f x)).indicator (fun _ => (1 : ℝ)) r *
(Set.Iic (g x)).indicator (fun _ => (1 : ℝ)) t := by
simp only [K,Set.indicator,Set.mem_Iic]; split_ifs <;> simp_all
simp_rw [he,integral_const_mul,integral_mul_const]
have hv (a : ℝ) (ha : 0 ≤ a ∧ a ≤ C) :
(∫ r, (Set.Iic a).indicator (fun _ => (1 : ℝ)) r ∂ν) = a := by
rw [integral_indicator measurableSet_Iic]
simp only [integral_const,smul_eq_mul,mul_one,measureReal_def]
dsimp [ν]
rw [Measure.restrict_apply MeasurableSet.univ,Set.univ_inter,
Measure.restrict_apply measurableSet_Iic]
have he : Set.Iic a ∩ Set.Ioc 0 C = Set.Ioc 0 a := by
ext r; simp only [Set.mem_inter_iff,Set.mem_Iic,Set.mem_Ioc]
constructor
· intro h; exact ⟨h.2.1,h.1⟩
· intro h; exact ⟨h.2,h.1,h.2.trans ha.2⟩
rw [he,Real.volume_Ioc,sub_zero,ENNReal.toReal_ofReal ha.1]
rw [hv _ (hfb x),hv _ (hgb x)]
calc
_ = ∫ x, ∫ p, K (p,x) ∂ν.prod ν ∂μ := by simp_rw [hpoint]
_ = ∫ p, ∫ x, K (p,x) ∂μ ∂ν.prod ν := (integral_integral_swap hKi).symm
_ = ∫ r, ∫ t, ∫ x, K ((r,t),x) ∂μ ∂ν ∂ν :=
integral_prod _ hKi.integral_prod_left
_ = _ := by
apply integral_congr_ae
filter_upwards [] with r
apply integral_congr_ae
filter_upwards [] with t
have he : (fun x => K ((r,t),x)) =
{x | r ≤ f x ∧ t ≤ g x}.indicator (fun _ => (1 : ℝ)) := by funext x; simp [K,Set.indicator]
have hm : MeasurableSet {x | r ≤ f x ∧ t ≤ g x} :=
(measurableSet_le measurable_const hf).inter (measurableSet_le measurable_const hg)
rw [he,integral_indicator hm]
simp [measureReal_def]
end
end FiniteFactor
end OAI
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