File size: 4,763 Bytes
b317718 | 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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import Mathlib
namespace OAI
namespace Problem367
open MeasureTheory ProbabilityTheory
open scoped ENNReal
variable {ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
variable (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
theorem indepFun_of_disjoint_finite_blocks (s t : Finset ι) (hst : Disjoint s t)
{α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(f : (∀ i : s, X i) → α) (g : (∀ i : t, X i) → β)
(hf : Measurable f) (hg : Measurable g) :
IndepFun (fun U : ∀ i, X i => f (fun i => U i))
(fun U : ∀ i, X i => g (fun i => U i)) (Measure.infinitePi μ) := by
have hind : iIndepFun (fun i (U : ∀ i, X i) => U i) (Measure.infinitePi μ) :=
iIndepFun_infinitePi (fun _ => measurable_id)
exact (hind.indepFun_finset s t hst (fun i => measurable_pi_apply i)).comp hf hg
theorem integral_mul_eq_of_disjoint_finite_blocks (s t : Finset ι) (hst : Disjoint s t)
(f : (∀ i : s, X i) → ℝ) (g : (∀ i : t, X i) → ℝ)
(hf : Measurable f) (hg : Measurable g) :
(∫ U, f (fun i => U i) * g (fun i => U i) ∂Measure.infinitePi μ) =
(∫ U, f (fun i => U i) ∂Measure.infinitePi μ) *
(∫ U, g (fun i => U i) ∂Measure.infinitePi μ) := by
apply (indepFun_of_disjoint_finite_blocks μ s t hst f g hf hg).integral_fun_mul_eq_mul_integral
all_goals exact Measurable.aestronglyMeasurable (by fun_prop)
theorem integral_mul_eq_zero_of_disjoint_finite_blocks (s t : Finset ι)
(hst : Disjoint s t) (f : (∀ i : s, X i) → ℝ) (g : (∀ i : t, X i) → ℝ)
(hf : Measurable f) (hg : Measurable g)
(hzero : (∫ U, f (fun i => U i) ∂Measure.infinitePi μ) = 0) :
(∫ U, f (fun i => U i) * g (fun i => U i) ∂Measure.infinitePi μ) = 0 := by
rw [integral_mul_eq_of_disjoint_finite_blocks μ s t hst f g hf hg, hzero, zero_mul]
theorem integral_mul_eq_zero_of_ae_finite_blocks (s t : Finset ι)
(hst : Disjoint s t) (f g : (∀ i, X i) → ℝ)
(F : (∀ i : s, X i) → ℝ) (G : (∀ i : t, X i) → ℝ)
(hF : Measurable F) (hG : Measurable G)
(hf : f =ᵐ[Measure.infinitePi μ] fun U => F (fun i => U i))
(hg : g =ᵐ[Measure.infinitePi μ] fun U => G (fun i => U i))
(hfm : AEStronglyMeasurable f (Measure.infinitePi μ))
(hgm : AEStronglyMeasurable g (Measure.infinitePi μ))
(hzero : (∫ U, f U ∂Measure.infinitePi μ) = 0) :
(∫ U, f U * g U ∂Measure.infinitePi μ) = 0 := by
have hind := (indepFun_of_disjoint_finite_blocks μ s t hst F G hF hG).congr hf.symm hg.symm
rw [hind.integral_fun_mul_eq_mul_integral hfm hgm, hzero, zero_mul]
theorem inner_eq_zero_of_disjoint_finite_blocks (s t : Finset ι)
(hst : Disjoint s t) (f g : Lp ℝ 2 (Measure.infinitePi μ))
(F : (∀ i : s, X i) → ℝ) (G : (∀ i : t, X i) → ℝ)
(hF : Measurable F) (hG : Measurable G)
(hf : f =ᵐ[Measure.infinitePi μ] fun U => F (fun i => U i))
(hg : g =ᵐ[Measure.infinitePi μ] fun U => G (fun i => U i))
(hzero : (∫ U, f U ∂Measure.infinitePi μ) = 0) :
inner ℝ f g = 0 := by
rw [L2.inner_def]
simpa [RCLike.inner_apply, mul_comm] using
integral_mul_eq_zero_of_ae_finite_blocks μ s t hst f g F G hF hG hf hg
(Lp.aestronglyMeasurable f) (Lp.aestronglyMeasurable g) hzero
section Reindex
variable {A : Type*} [MeasurableSpace A] (ν : Measure A) [IsProbabilityMeasure ν]
theorem indepFun_finite_blocks_reindex (s t : Finset ι) (e : ι ≃ ι)
(hst : Disjoint s (t.map e.toEmbedding))
(f : (s → A) → ℝ) (g : (t → A) → ℝ) (hf : Measurable f) (hg : Measurable g) :
IndepFun (fun U : ι → A => f (fun i => U i))
(fun U : ι → A => g (fun i => U (e i))) (Measure.infinitePi fun _ : ι => ν) := by
let G : (t.map e.toEmbedding → A) → ℝ := fun U =>
g (fun i => U ⟨e i, Finset.mem_map.2 ⟨i, i.property, rfl⟩⟩)
have hG : Measurable G := by
apply hg.comp
fun_prop
simpa [G] using indepFun_of_disjoint_finite_blocks (fun _ : ι => ν)
s (t.map e.toEmbedding) hst f G hf hG
theorem integral_mul_reindex_eq_zero_of_disjoint (s t : Finset ι) (e : ι ≃ ι)
(hst : Disjoint s (t.map e.toEmbedding))
(f : (s → A) → ℝ) (g : (t → A) → ℝ) (hf : Measurable f) (hg : Measurable g)
(hzero : (∫ U, f (fun i => U i) ∂Measure.infinitePi (fun _ : ι => ν)) = 0) :
(∫ U, f (fun i => U i) * g (fun i => U (e i))
∂Measure.infinitePi (fun _ : ι => ν)) = 0 := by
have hind := indepFun_finite_blocks_reindex ν s t e hst f g hf hg
rw [hind.integral_fun_mul_eq_mul_integral
(Measurable.aestronglyMeasurable (by fun_prop))
(Measurable.aestronglyMeasurable (by fun_prop)), hzero, zero_mul]
end Reindex
end Problem367
end OAI
|