Bounded Multipliers Preserve Finite Squared Energy

Proved

LevyStochCalc.Brownian.Ito.lintegral_sq_bounded_mul_lt_top_readable

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

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

This technical statement shows that a bounded multiplier preserves local square-integrability. Let $H:Ω×ℝ→ℝ$ be such that, for every $T>0$ , its iterated squared ENNReal energy is strictly below $⊤$ . Let $g:Ω×ℝ→ℝ$ and $K_g∈ℝ$ satisfy $0≤K_g$ and $|g(ω,s)|≤K_g$ for all $ω,s$ .

Then, for every $T>0$ ,

\int_Ω\int_{[0,T]}\|g(ω,s)H(ω,s)\|^2\,ds\,dP<⊤.

The integrals take values in $[0,∞]$ before the strict inequality is imposed. No probability-measure assumption is required for $P$ in this declaration. This lemma supplies the energy estimate needed when multiplying an integrand by bounded coefficients.

Original Lean declaration header

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

theorem lintegral_sq_bounded_mul_lt_top
    (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {g : Ω → ℝ → ℝ} {Kg : ℝ} (hKg0 : 0 ≤ Kg) (hKg : ∀ ω s, |g ω s| ≤ Kg) :
    ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖g ω s * H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤

Module navigation

Module: LevyStochCalc/Brownian/ItoMartingaleRiemann.

Source excerpt, lines 51–56 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoMartingaleRiemann

/-
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/ItoMartingaleRiemann.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {H : Ω → ℝ → ℝ}
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.lintegral_sq_bounded_mul_lt_top_readable
    (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {g : Ω → ℝ → ℝ} {Kg : ℝ} (hKg0 : 0 ≤ Kg) (hKg : ∀ ω s, |g ω s| ≤ Kg) :
    ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖g ω s * H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoMartingaleRiemann.lean#L49-L79 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoMartingaleRiemann`, declaration `LevyStochCalc.Brownian.Ito.lintegral_sq_bounded_mul_lt_top`; 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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