| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| variable {Ω : Type*} [MeasurableSpace Ω] | |
| lemma sq_integral_mul_le_integral_sq_of_abs_le_one {μ : Measure Ω} [IsProbabilityMeasure μ] | |
| {g f : Ω → ℝ} (hg : MemLp g 2 μ) (hf : AEStronglyMeasurable f μ) | |
| (hfBound : ∀ᵐ w ∂μ, |f w| ≤ 1) : | |
| (∫ w, g w * f w ∂μ) ^ 2 ≤ ∫ w, g w ^ 2 ∂μ := by | |
| have hgi := hg.integrable one_le_two | |
| have hfNorm : ∀ᵐ w ∂μ, ‖f w‖ ≤ 1 := by simpa only [Real.norm_eq_abs] using hfBound | |
| have hprod := hgi.mul_bdd hf hfNorm | |
| have habs : |∫ w, g w * f w ∂μ| ≤ ∫ w, |g w| ∂μ := by | |
| calc | |
| |∫ w, g w * f w ∂μ| ≤ ∫ w, |g w * f w| ∂μ := abs_integral_le_integral_abs | |
| _ ≤ ∫ w, |g w| ∂μ := by | |
| apply integral_mono_ae hprod.abs hgi.abs | |
| filter_upwards [hfBound] with w hw | |
| rw [abs_mul] | |
| simpa using mul_le_mul_of_nonneg_left hw (abs_nonneg (g w)) | |
| have hvar := variance_nonneg (fun w => |g w|) μ | |
| have hgAbs : MemLp (fun w => |g w|) 2 μ := by | |
| simpa only [Real.norm_eq_abs] using hg.norm | |
| rw [variance_eq_sub hgAbs] at hvar | |
| simp only [Pi.pow_apply, sq_abs] at hvar | |
| have hn : 0 ≤ ∫ w, |g w| ∂μ := integral_nonneg fun w => abs_nonneg _ | |
| have hs := (sq_le_sq₀ (abs_nonneg _) hn).mpr habs | |
| rw [sq_abs] at hs | |
| linarith | |
| lemma variance_ge_of_binary_tilt_moments {μ ν : Measure Ω} | |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| {g f : Ω → ℝ} {theta : ℝ} | |
| (hg : MemLp g 2 μ) (hgTilt : MemLp g 2 ν) | |
| (hf : AEStronglyMeasurable f μ) (hfBound : ∀ᵐ w ∂μ, |f w| ≤ 1) | |
| (hFirst : (∫ w, g w ∂ν) = theta * ∫ w, g w * f w ∂μ) | |
| (hSecond : (∫ w, g w ^ 2 ∂ν) = ∫ w, g w ^ 2 ∂μ) : | |
| (1 - theta ^ 2) * (∫ w, g w ^ 2 ∂μ) ≤ variance g ν := by | |
| have hcs := sq_integral_mul_le_integral_sq_of_abs_le_one hg hf hfBound | |
| have hmul := mul_le_mul_of_nonneg_left hcs (sq_nonneg theta) | |
| rw [variance_eq_sub hgTilt] | |
| change (1 - theta ^ 2) * (∫ w, g w ^ 2 ∂μ) ≤ | |
| (∫ w, g w ^ 2 ∂ν) - (∫ w, g w ∂ν) ^ 2 | |
| rw [hFirst, hSecond] | |
| nlinarith | |
| end Problem367 | |
| end | |
| end OAI | |