Second-Moment Bound for a Finite Sum

Proved

LevyStochCalc.Brownian.Ito.integrable_sq_sum_and_le_readable

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

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

Let $\iota$ be a finite type, let $f_i:\Omega\to\mathbb R$ be measurable, and suppose each $f_i^2$ is integrable under $P$ . Then the square of the finite sum is integrable and

\int\left(\sum_{i\in\iota}f_i\right)^2dP \le |\iota|\sum_{i\in\iota}\int f_i^2\,dP.

The cardinality $|\iota|$ is interpreted as a real number in the inequality. This is a finite-sum Cauchy–Schwarz estimate used for coordinatewise Itô-process moment bounds.

Original Lean declaration header

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

theorem integrable_sq_sum_and_le {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => (f i ω) ^ 2) P) :
    Integrable (fun ω => (∑ i, f i ω) ^ 2) P
      ∧ ∫ ω, (∑ i, f i ω) ^ 2 ∂P
        ≤ (Fintype.card ι : ℝ) * ∑ i, ∫ ω, (f i ω) ^ 2 ∂P

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

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Source excerpt, lines 102–106 · 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_sum_and_le_readable {f : ι → Ω → ℝ} (hfm : ∀ i, Measurable (f i))
    (hfi : ∀ i, Integrable (fun ω => (f i ω) ^ 2) P) :
    Integrable (fun ω => (∑ i, f i ω) ^ 2) P
      ∧ ∫ ω, (∑ i, f i ω) ^ 2 ∂P
        ≤ (Fintype.card ι : ℝ) * ∑ i, ∫ ω, (f i ω) ^ 2 ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L100-L125 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_sq_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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