| import Mathlib.MeasureTheory.Integral.Bochner.Basic |
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| namespace OAI |
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| /-! # A common density gives a strict oscillation bound for weighted means |
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| This is the scalar estimate for the derivative of two positive-kernel |
| log transforms. All densities and products are integrated explicitly. |
| -/ |
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| open Set Filter MeasureTheory |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {Y : Type*} [MeasurableSpace Y] (μ : Measure Y) |
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| 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] |
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| 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] |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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