| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| namespace Problem367 |
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| theorem sq_integral_le_integral_sq {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ} (hf : MemLp f 2 μ) : |
| (∫ x, f x ∂μ)^2 ≤ ∫ x, (f x)^2 ∂μ := by |
| have h := variance_nonneg f μ |
| rw [variance_eq_sub hf] at h |
| simpa only [Pi.pow_apply] using (sub_nonneg.mp h) |
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| theorem memLp_integral_noise {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| {μ : Measure Ω} {ν : Measure A} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {f : Ω × A → ℝ} (hm : Measurable f) (hf : MemLp f 2 (μ.prod ν)) : |
| MemLp (fun x => ∫ y, f (x,y) ∂ν) 2 μ ∧ |
| (∫ x, (∫ y, f (x,y) ∂ν)^2 ∂μ) ≤ ∫ z, (f z)^2 ∂μ.prod ν := by |
| have havg : Measurable (fun x => ∫ y, f (x,y) ∂ν) := |
| hm.stronglyMeasurable.integral_prod_right'.measurable |
| have hbound : ∀ᵐ x ∂μ, (∫ y, f (x,y) ∂ν)^2 ≤ ∫ y, (f (x,y))^2 ∂ν := by |
| filter_upwards [hf.integrable_sq.prod_right_ae] with x hx |
| apply sq_integral_le_integral_sq |
| exact (memLp_two_iff_integrable_sq |
| (hm.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable).2 hx |
| have hint : Integrable (fun x => (∫ y, f (x,y) ∂ν)^2) μ := by |
| apply hf.integrable_sq.integral_prod_left.mono' (havg.pow_const 2).aestronglyMeasurable |
| filter_upwards [hbound] with x hx |
| simpa only [Real.norm_eq_abs, abs_sq] using hx |
| refine ⟨(memLp_two_iff_integrable_sq havg.aestronglyMeasurable).2 hint, ?_⟩ |
| rw [integral_prod _ hf.integrable_sq] |
| exact integral_mono_ae hint hf.integrable_sq.integral_prod_left hbound |
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| end Problem367 |
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| end |
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| end OAI |
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