Integrability and Moment Bound for a Finite Sum

Proved

LevyStochCalc.Brownian.Ito.integrable_abs_sum_and_le_readable

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

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

This is the finite-sum version of the absolute-moment estimate.

Let $\iota$ be a finite type, let $f_i:\Omega\to\mathbb R$ be measurable for each $i\in\iota$ , and assume $|f_i|$ is integrable for every $i$ . Then $\left|\sum_i f_i\right|$ is integrable and

\int\left|\sum_{i\in\iota}f_i(\omega)\right|\,dP \le \sum_{i\in\iota}\int|f_i(\omega)|\,dP.

This estimate is used to control finite collections of drift and stochastic-integral increments.

Original Lean declaration header

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

theorem integrable_abs_sum_and_le {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => |f i ω|) P) :
    Integrable (fun ω => |∑ i, f i ω|) P
      ∧ ∫ ω, |∑ i, f i ω| ∂P ≤ ∑ i, ∫ ω, |f i ω| ∂P

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

Next: integrable_sq_sum_and_le.

Source excerpt, lines 82–85 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoProcess

/-
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/VectorItoProcess.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} {ι : Type*} [Fintype ι]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integrable_abs_sum_and_le_readable {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => |f i ω|) P) :
    Integrable (fun ω => |∑ i, f i ω|) P
      ∧ ∫ ω, |∑ i, f i ω| ∂P ≤ ∑ i, ∫ ω, |f i ω| ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L80-L98 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_abs_sum_and_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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