Progressive Measurability Of The Global Patched Path

Proved

LevyStochCalc.Ito.Picard.progressivelyMeasurable_globalPatch_readable

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

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

This is a technical measurability lemma. Let $\mathcal F$ be a filtration and let $Y_k:\mathbb R\to\Omega\to(\operatorname{Fin}(n)\to\mathbb R)$ be a countable family such that every coordinate of every $Y_k$ is progressively measurable. Then every coordinate of the glued path globalPatch Y is progressively measurable:

\forall i\in\operatorname{Fin}(n),\quad (\omega,s)\mapsto\operatorname{globalPatch}(Y,s,\omega,i)\text{ is progressively measurable}.

The result supports construction of a global process by selecting, below each time, one of finitely many window paths.

Original Lean declaration header

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

theorem progressivelyMeasurable_globalPatch
    {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›} {Y : ℕ → ℝ → Ω → (Fin n → ℝ)}
    (hY : ∀ k, ∀ i : Fin n, Probability.ProgressivelyMeasurable ℱ fun ω s => Y k s ω i)
    (i : Fin n) :
    Probability.ProgressivelyMeasurable ℱ fun ω s => globalPatch Y s ω i

Module navigation

Module: LevyStochCalc/Ito/PicardGlobal.

Definition modules: LevyStochCalc/Probability/Progressive.

Previous: ae_all_eq_of_rightCont · Next: ae_all_windows_agree.

Source excerpt, lines 102–106 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_PicardGlobal

/-
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/Ito/PicardGlobal.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Probability
universe u v
variable {Ω : Type u} [MeasurableSpace Ω] {E : Type v} [MeasurableSpace E]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n d : ℕ}
omit [MeasurableSpace E] [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Ito.Picard.progressivelyMeasurable_globalPatch_readable
    {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›} {Y : ℕ → ℝ → Ω → (Fin n → ℝ)}
    (hY : ∀ k, ∀ i : Fin n, Probability.ProgressivelyMeasurable ℱ fun ω s => Y k s ω i)
    (i : Fin n) :
    Probability.ProgressivelyMeasurable ℱ fun ω s => globalPatch Y s ω i := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardGlobal.lean#L99-L134 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardGlobal`, declaration `LevyStochCalc.Ito.Picard.progressivelyMeasurable_globalPatch`; 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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