Progressive Measurability of a Weighted Window Indicator

Proved

LevyStochCalc.Brownian.Ito.progressivelyMeasurable_mul_indIoc_readable

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

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

This proves progressive measurability for a weighted time window.

Let $\mathcal F$ be a filtration, let $0\le a<b$ , and let $V:\Omega\to\mathbb R$ satisfy:

$V$ is bounded;
$V$ is measurable;
$V$ is strongly measurable with respect to $\mathcal F_a$ .

Then the process $(\omega,s)\mapsto V(\omega)\mathbf 1_{(a,b]}(s)$ is progressively measurable with respect to $\mathcal F$ :

\operatorname{ProgressivelyMeasurable}_{\mathcal F}\bigl((\omega,s)\mapsto V(\omega)\mathbf 1_{(a,b]}(s)\bigr).

The lemma supplies the measurability condition needed to use a past weight as an Itô-integral integrand.

Original Lean declaration header

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

theorem progressivelyMeasurable_mul_indIoc (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) {a b : ℝ}
    (ha : 0 ≤ a) (hab : a < b) {V : Ω → ℝ} (hVb : ∃ M : ℝ, ∀ ω, |V ω| ≤ M)
    (hVm : Measurable V) (hVa : @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ a) V) :
    Probability.ProgressivelyMeasurable ℱ fun ω s => V ω * indIoc Ω a b ω s

Module navigation

Module: LevyStochCalc/Brownian/ItoPullOut.

Definition modules: LevyStochCalc/Brownian/ItoIncrement, LevyStochCalc/Probability/Progressive.

Supporting theorems: stepIocMul_eval, stepIocMul₀_eval.

Previous: stepIocMul₀_eval · Next: mul_stochasticIntegralBrownian_indIoc.

Source excerpt, lines 149–152 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoPullOut

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Modified 2026-10-12 by the KLS formalization project: ported to Lean v4.33.0 and
-- Mathlib db584cd. The changes are listed in vendor/README.md and vendor/port.patch.
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ItoPullOut.lean; Apache-2.0; locally patched, see formalization/vendor/port.patch; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


theorem LevyStochCalc.Brownian.Ito.progressivelyMeasurable_mul_indIoc_readable (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) {a b : ℝ}
    (ha : 0 ≤ a) (hab : a < b) {V : Ω → ℝ} (hVb : ∃ M : ℝ, ∀ ω, |V ω| ≤ M)
    (hVm : Measurable V) (hVa : @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ a) V) :
    Probability.ProgressivelyMeasurable ℱ fun ω s => V ω * indIoc Ω a b ω s := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoPullOut.lean (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoPullOut`, declaration `LevyStochCalc.Brownian.Ito.progressivelyMeasurable_mul_indIoc`; the repository is private at upload time); vendored copy patched by formalization/vendor/port.patch (project file formalization/vendor/LevyStochCalc/LevyStochCalc/Brownian/ItoPullOut.lean, lines 147-165). 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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