| import Mathlib |
| import OAI.Computability.VertexCover.Fourier.SourceOccurrences |
| import OAI.Computability.VertexCover.Analysis.PoincareOne |
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| namespace OAI |
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| namespace VertexCover.Product |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| variable (μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
|
|
| theorem bounded_memLp (f : X → ℝ) (hf : Measurable f) {C : ℝ} |
| (hb : ∀ x, ‖f x‖ ≤ C) : MemLp f 2 μ := |
| MemLp.of_bound hf.aestronglyMeasurable C (Filter.Eventually.of_forall hb) |
|
|
| theorem bounded_integral_norm (f : X → ℝ) {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C) : |
| ‖∫ x, f x ∂μ‖ ≤ C := by |
| simpa only [probReal_univ, mul_one] using |
| norm_integral_le_of_norm_le_const (μ := μ) (Filter.Eventually.of_forall hb) |
|
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| theorem mean_measurable (f : X → Y → ℝ) (hf : Measurable (Function.uncurry f)) : |
| Measurable (fun x => ∫ y, f x y ∂ν) := |
| hf.stronglyMeasurable.integral_prod_right.measurable |
|
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| theorem mean_memLp (f : X → Y → ℝ) (hf : Measurable (Function.uncurry f)) {C : ℝ} |
| (hb : ∀ x y, ‖f x y‖ ≤ C) : MemLp (fun x => ∫ y, f x y ∂ν) 2 μ := |
| bounded_memLp μ _ (mean_measurable ν f hf) (fun x => bounded_integral_norm ν _ (hb x)) |
|
|
| theorem variance_product (f : X → Y → ℝ) (hf : Measurable (Function.uncurry f)) {C : ℝ} |
| (hb : ∀ x y, ‖f x y‖ ≤ C) : |
| variance (Function.uncurry f) (μ.prod ν) = |
| (∫ x, variance (f x) ν ∂μ) + variance (fun x => ∫ y, f x y ∂ν) μ := by |
| have hp := bounded_memLp (μ.prod ν) _ hf (fun p => hb p.1 p.2) |
| have hs : ∀ x, MemLp (f x) 2 ν := fun x => |
| bounded_memLp ν _ (hf.comp (measurable_const.prodMk measurable_id)) (hb x) |
| have hm := mean_memLp μ ν f hf hb |
| simp_rw [variance_eq_sub (hs _)] |
| rw [variance_eq_sub hp, variance_eq_sub hm] |
| simp only [Pi.pow_apply] |
| dsimp only [Function.uncurry] at hp ⊢ |
| have hi : Integrable (fun x => ∫ y, (f x y)^2 ∂ν) μ := hp.integrable_sq.integral_prod_left |
| have he2 : (∫ p : X × Y, (f p.1 p.2)^2 ∂μ.prod ν) = |
| ∫ x, ∫ y, (f x y)^2 ∂ν ∂μ := integral_prod _ hp.integrable_sq |
| have he1 : (∫ p : X × Y, f p.1 p.2 ∂μ.prod ν) = |
| ∫ x, ∫ y, f x y ∂ν ∂μ := integral_prod _ (hp.integrable (by norm_num)) |
| rw [integral_sub hi hm.integrable_sq, he2, he1] |
| ring |
|
|
| theorem section_variance_integrable (f : X → Y → ℝ) |
| (hf : Measurable (Function.uncurry f)) {C : ℝ} (hb : ∀ x y, ‖f x y‖ ≤ C) : |
| Integrable (fun x => variance (f x) ν) μ := by |
| have hp := bounded_memLp (μ.prod ν) _ hf (fun p => hb p.1 p.2) |
| have hs : ∀ x, MemLp (f x) 2 ν := fun x => |
| bounded_memLp ν _ (hf.comp (measurable_const.prodMk measurable_id)) (hb x) |
| have hm := mean_memLp μ ν f hf hb |
| simp_rw [variance_eq_sub (hs _)] |
| exact hp.integrable_sq.integral_prod_left.sub hm.integrable_sq |
|
|
| theorem variance_mean_le (f : X → Y → ℝ) (hf : Measurable (Function.uncurry f)) {C : ℝ} |
| (hb : ∀ x y, ‖f x y‖ ≤ C) : |
| variance (fun x => ∫ y, f x y ∂ν) μ ≤ ∫ y, variance (fun x => f x y) μ ∂ν := by |
| let g : Y → ℝ := fun y => ∫ x, f x y ∂μ |
| let D : X → Y → ℝ := fun x y => f x y - g y |
| have hg : Measurable g := mean_measurable μ (fun y x => f x y) (hf.comp measurable_swap) |
| have hgb : ∀ y, ‖g y‖ ≤ C := fun y => bounded_integral_norm μ _ (fun x => hb x y) |
| have hd : Measurable (Function.uncurry D) := hf.sub (hg.comp measurable_snd) |
| have hdb : ∀ x y, ‖D x y‖ ≤ 2*C := fun x y => by |
| exact (norm_sub_le _ _).trans (by linarith [hb x y, hgb y]) |
| have hp := bounded_memLp (μ.prod ν) _ hd (fun p => hdb p.1 p.2) |
| have hs : ∀ x, MemLp (D x) 2 ν := fun x => |
| bounded_memLp ν _ (hd.comp (measurable_const.prodMk measurable_id)) (hdb x) |
| have hm := mean_memLp μ ν f hf hb |
| have hfp := bounded_memLp (μ.prod ν) _ hf (fun p => hb p.1 p.2) |
| have hfx : ∀ x, Integrable (f x) ν := fun x => |
| (bounded_memLp ν _ (hf.comp (measurable_const.prodMk measurable_id)) (hb x)).integrable (by norm_num) |
| have hgi := (bounded_memLp ν g hg hgb).integrable (by norm_num : (1 : ENNReal) ≤ 2) |
| have hei : ∫ y, g y ∂ν = ∫ x, ∫ y, f x y ∂ν ∂μ := by |
| exact (integral_integral_swap (hfp.integrable (by norm_num))).symm |
| have he : ∀ x, (∫ y, f x y ∂ν) - ∫ x, ∫ y, f x y ∂ν ∂μ = ∫ y, D x y ∂ν := by |
| intro x |
| dsimp only [D] |
| rw [integral_sub (hfx x) hgi, hei] |
| rw [variance_eq_integral hm.aemeasurable] |
| simp_rw [he] |
| calc |
| _ ≤ ∫ x, ∫ y, (D x y)^2 ∂ν ∂μ := by |
| apply integral_mono |
| · exact (mean_memLp μ ν D hd hdb).integrable_sq |
| · exact hp.integrable_sq.integral_prod_left |
| · intro x |
| simpa only [probReal_univ, one_mul] using Cube.sq_integral_le_measure_mul ν (hs x) |
| _ = ∫ y, ∫ x, (D x y)^2 ∂μ ∂ν := integral_integral_swap hp.integrable_sq |
| _ = _ := by |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall (fun y => |
| (variance_eq_integral (hf.comp (measurable_id.prodMk measurable_const)).aemeasurable).symm) |
|
|
| theorem variance_product_le (f : X → Y → ℝ) (hf : Measurable (Function.uncurry f)) {C : ℝ} |
| (hb : ∀ x y, ‖f x y‖ ≤ C) : |
| variance (Function.uncurry f) (μ.prod ν) ≤ |
| (∫ x, variance (f x) ν ∂μ) + ∫ y, variance (fun x => f x y) μ ∂ν := by |
| rw [variance_product μ ν f hf hb] |
| exact add_le_add le_rfl (variance_mean_le μ ν f hf hb) |
|
|
| end VertexCover.Product |
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| end OAI |
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