Second-Moment Bound for a Sum of Two Functions

Proved

LevyStochCalc.Brownian.Ito.integrable_sq_add_and_le_readable

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

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

Let $A,B:\Omega\to\mathbb R$ be measurable, with $A^2$ and $B^2$ integrable under a measure $P$ . If

\int A^2\,dP\le a,\qquad \int B^2\,dP\le b,

then $(A+B)^2$ is integrable and

\int(A+B)^2\,dP\le2a+2b.

This elementary estimate is useful for combining drift and martingale contributions while retaining explicit second-moment bounds.

Original Lean declaration header

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

theorem integrable_sq_add_and_le {A B : Ω → ℝ} (hAm : Measurable A) (hBm : Measurable B)
    (hA : Integrable (fun ω => (A ω) ^ 2) P) (hB : Integrable (fun ω => (B ω) ^ 2) P)
    {a b : ℝ} (hAle : ∫ ω, (A ω) ^ 2 ∂P ≤ a) (hBle : ∫ ω, (B ω) ^ 2 ∂P ≤ b) :
    Integrable (fun ω => (A ω + B ω) ^ 2) P
      ∧ ∫ ω, (A ω + B ω) ^ 2 ∂P ≤ 2 * a + 2 * b

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

Previous: integrable_abs_sum_pow_three · Next: integrable_abs_add_and_le.

Source excerpt, lines 147–151 · 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_sq_add_and_le_readable {A B : Ω → ℝ} (hAm : Measurable A) (hBm : Measurable B)
    (hA : Integrable (fun ω => (A ω) ^ 2) P) (hB : Integrable (fun ω => (B ω) ^ 2) P)
    {a b : ℝ} (hAle : ∫ ω, (A ω) ^ 2 ∂P ≤ a) (hBle : ∫ ω, (B ω) ^ 2 ∂P ≤ b) :
    Integrable (fun ω => (A ω + B ω) ^ 2) P
      ∧ ∫ ω, (A ω + B ω) ^ 2 ∂P ≤ 2 * a + 2 * b := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L145-L167 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_sq_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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