File size: 2,210 Bytes
f61e547 | 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 | 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
|