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b37e936 | 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 | import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
/-! # A common density gives a strict oscillation bound for weighted means
This is the scalar estimate for the derivative of two positive-kernel
log transforms. All densities and products are integrated explicitly.
-/
open Set Filter MeasureTheory
noncomputable section
namespace RegularPointCoordinates
variable {Y : Type*} [MeasurableSpace Y] (μ : Measure Y)
theorem weighted_mean_minorization {r w f : Y → ℝ} {q l u : ℝ}
(hr : Integrable r μ) (hw : Integrable w μ)
(hfr : Integrable (fun x => f x*r x) μ) (hfw : Integrable (fun x => f x*w x) μ)
(hr1 : (∫ x, r x ∂μ)=1) (hw1 : (∫ x, w x ∂μ)=1)
(hmin : ∀ᵐ x ∂μ, q*r x ≤ w x) (hf : ∀ᵐ x ∂μ, l ≤ f x ∧ f x ≤ u) :
q*(∫ x, f x*r x ∂μ)+(1-q)*l ≤ ∫ x, f x*w x ∂μ ∧
(∫ x, f x*w x ∂μ) ≤ q*(∫ x, f x*r x ∂μ)+(1-q)*u := by
let d := fun x => w x-q*r x
have hd : Integrable d μ := hw.sub (hr.const_mul q)
have hfd : Integrable (fun x => f x*d x) μ := by
convert hfw.sub (hfr.const_mul q) using 1
funext x
change f x * (w x - q * r x) = f x * w x - q * (f x * r x)
ring
have hdint : (∫ x, d x ∂μ)=1-q := by
rw [integral_sub hw (hr.const_mul q),integral_const_mul,hr1,hw1,mul_one]
have hfdint : (∫ x, f x*d x ∂μ) = (∫ x, f x*w x ∂μ)-q*(∫ x, f x*r x ∂μ) := by
have he : (fun x => f x*d x) = (fun x => f x*w x - q*(f x*r x)) := by
funext x
dsimp only [d]
ring
rw [he,integral_sub hfw (hfr.const_mul q),integral_const_mul]
have hlo : (∫ x, l*d x ∂μ) ≤ ∫ x, f x*d x ∂μ := by
apply integral_mono_ae (hd.const_mul l) hfd
filter_upwards [hmin,hf] with x hx hf
exact mul_le_mul_of_nonneg_right hf.1 (sub_nonneg.mpr hx)
have hhi : (∫ x, f x*d x ∂μ) ≤ ∫ x, u*d x ∂μ := by
apply integral_mono_ae hfd (hd.const_mul u)
filter_upwards [hmin,hf] with x hx hf
exact mul_le_mul_of_nonneg_right hf.2 (sub_nonneg.mpr hx)
rw [integral_const_mul,hdint,hfdint] at hlo hhi
constructor <;> linarith only [hlo,hhi]
theorem common_density_mean_difference {r w v f : Y → ℝ} {q l u : ℝ}
(hr : Integrable r μ) (hw : Integrable w μ) (hv : Integrable v μ)
(hfr : Integrable (fun x => f x*r x) μ) (hfw : Integrable (fun x => f x*w x) μ)
(hfv : Integrable (fun x => f x*v x) μ)
(hr1 : (∫ x, r x ∂μ)=1) (hw1 : (∫ x, w x ∂μ)=1) (hv1 : (∫ x, v x ∂μ)=1)
(hwmin : ∀ᵐ x ∂μ, q*r x ≤ w x) (hvmin : ∀ᵐ x ∂μ, q*r x ≤ v x)
(hf : ∀ᵐ x ∂μ, l ≤ f x ∧ f x ≤ u) :
|(∫ x, f x*w x ∂μ)-(∫ x, f x*v x ∂μ)| ≤ (1-q)*(u-l) := by
have hW := weighted_mean_minorization μ hr hw hfr hfw hr1 hw1 hwmin hf
have hV := weighted_mean_minorization μ hr hv hfr hfv hr1 hv1 hvmin hf
apply abs_le.mpr
constructor <;> nlinarith only [hW.1,hW.2,hV.1,hV.2]
end RegularPointCoordinates
end
end OAI
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