| import Mathlib |
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| namespace OAI |
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| namespace Problem367 |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
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| variable {ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)] |
| variable (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)] |
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| 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 |
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| 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) |
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| 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] |
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| 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] |
|
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| 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 |
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| section Reindex |
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| variable {A : Type*} [MeasurableSpace A] (ν : Measure A) [IsProbabilityMeasure ν] |
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| 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 |
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| 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] |
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| end Reindex |
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| end Problem367 |
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| end OAI |
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