First-Moment Bound for a Weighted Finite Cell Sum

Proved

LevyStochCalc.Brownian.Ito.integral_abs_weighted_cellSum_le_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This is a finite-sum $L^1$ estimate for weighted random variables.

Let $P$ be a measure on $\Omega$ , let $w_i,R_i:\Omega\to\mathbb R$ be measurable for every $i\in\mathbb N$ , and let $K_w\ge0$ . Assume $|w_i(\omega)|\le K_w$ , each $|R_i|$ is integrable, and for some $\rho\in\mathbb R$ ,

\int |R_i|\,dP\le\rho\quad\text{for every }i.

Then the finite sum $\sum_{i<N}w_iR_i$ is integrable and

\int\left|\sum_{i<N}w_iR_i\right|dP\le N K_w\rho.

This auxiliary lemma packages the triangle inequality and a common first-moment bound for cell-sum estimates; it does not require that $P$ be a probability measure.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem integral_abs_weighted_cellSum_le (N : ℕ) (w R : ℕ → Ω → ℝ)
    (hwm : ∀ i, Measurable (w i)) (hRm : ∀ i, Measurable (R i))
    {Kw : ℝ} (hKw0 : 0 ≤ Kw) (hwb : ∀ (i : ℕ) (ω : Ω), |w i ω| ≤ Kw)
    (hRint : ∀ i, Integrable (fun ω => |R i ω|) P)
    {ρ : ℝ} (hRle : ∀ i, ∫ ω, |R i ω| ∂P ≤ ρ) :
    Integrable (fun ω => ∑ i ∈ Finset.range N, w i ω * R i ω) P
      ∧ ∫ ω, |∑ i ∈ Finset.range N, w i ω * R i ω| ∂P ≤ (N : ℝ) * (Kw * ρ)

Module navigation

Module: LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.

Next: IsVectorItoVersion.integral_abs_sum_drift_sq_le.

Source excerpt, lines 46–52 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoQuadVarRiemannTermBounds

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_weighted_cellSum_le_readable (N : ℕ) (w R : ℕ → Ω → ℝ)
    (hwm : ∀ i, Measurable (w i)) (hRm : ∀ i, Measurable (R i))
    {Kw : ℝ} (hKw0 : 0 ≤ Kw) (hwb : ∀ (i : ℕ) (ω : Ω), |w i ω| ≤ Kw)
    (hRint : ∀ i, Integrable (fun ω => |R i ω|) P)
    {ρ : ℝ} (hRle : ∀ i, ∫ ω, |R i ω| ∂P ≤ ρ) :
    Integrable (fun ω => ∑ i ∈ Finset.range N, w i ω * R i ω) P
      ∧ ∫ ω, |∑ i ∈ Finset.range N, w i ω * R i ω| ∂P ≤ (N : ℝ) * (Kw * ρ) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.lean#L42-L77 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoQuadVarRiemannTermBounds`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_weighted_cellSum_le`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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