Integrability and Moment Bound for Two Summands

Proved

LevyStochCalc.Brownian.Ito.integrable_abs_add_and_le_readable

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

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

This is an integrability and first-moment estimate for a sum.

Let $A,B:\Omega\to\mathbb R$ be measurable. Assume $|A|$ and $|B|$ are integrable with respect to a measure $P$ . If real numbers $a,b$ satisfy

\int |A|\,dP\le a, \qquad \int |B|\,dP\le b,

then $|A+B|$ is integrable and

\int |A+B|\,dP\le a+b.

This is a reusable two-term estimate for moment bounds of process increments.

Original Lean declaration header

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

theorem integrable_abs_add_and_le {A B : Ω → ℝ} (hAm : Measurable A) (hBm : Measurable B)
    (hA : Integrable (fun ω => |A ω|) P) (hB : Integrable (fun ω => |B ω|) P)
    {a b : ℝ} (hAle : ∫ ω, |A ω| ∂P ≤ a) (hBle : ∫ ω, |B ω| ∂P ≤ b) :
    Integrable (fun ω => |A ω + B ω|) P ∧ ∫ ω, |A ω + B ω| ∂P ≤ a + b

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

Previous: integrable_sq_add_and_le · Next: integral_sq_vectorItoMartingale_sub_le.

Source excerpt, lines 171–174 · 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_add_and_le_readable {A B : Ω → ℝ} (hAm : Measurable A) (hBm : Measurable B)
    (hA : Integrable (fun ω => |A ω|) P) (hB : Integrable (fun ω => |B ω|) P)
    {a b : ℝ} (hAle : ∫ ω, |A ω| ∂P ≤ a) (hBle : ∫ ω, |B ω| ∂P ≤ b) :
    Integrable (fun ω => |A ω + B ω|) P ∧ ∫ ω, |A ω + B ω| ∂P ≤ a + b := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L169-L186 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_abs_add_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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