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import OAI.Computability.VertexCover.Fourier.SourceOccurrences
import OAI.Computability.VertexCover.Analysis.PoincareOne
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)
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
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
end
end
end
end
end
end
end
end
end
end
end
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end
end
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end
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end
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end
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end
end
end OAI
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