Integrability of the Cubed Absolute Value of a Finite Sum

Proved

LevyStochCalc.Brownian.Ito.integrable_abs_sum_pow_three_readable

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

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

This is a finite-sum integrability lemma.

Let $\iota$ be a finite index type, let $f_i:\Omega\to\mathbb R$ be measurable for every $i\in\iota$ , and assume that each $|f_i|^3$ is integrable with respect to a measure $P$ . Then

\omega\longmapsto\left|\sum_{i\in\iota}f_i(\omega)\right|^3

is integrable with respect to $P$ .

The result is independent of any probability assumption and is used to assemble third-moment estimates from finitely many coordinates or summands.

Original Lean declaration header

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

theorem integrable_abs_sum_pow_three {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => |f i ω| ^ 3) P) :
    Integrable (fun ω => |∑ i, f i ω| ^ 3) P

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

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Source excerpt, lines 128–130 · 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_pow_three_readable {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => |f i ω| ^ 3) P) :
    Integrable (fun ω => |∑ i, f i ω| ^ 3) P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L127-L143 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_abs_sum_pow_three`; 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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