| import OAI.Analysis.FiniteFactor.Sampling.Rectangle |
| import Mathlib.MeasureTheory.Integral.Layercake |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal |
| noncomputable section |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
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| end OAI |
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