Mutual Independence From Independence Of Strict Predecessors

Proved

LevyStochCalc.Brownian.iIndep_of_indep_biSup_lt_readable

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

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

Let $\iota$ be a linearly ordered type and let $m_i$ be sigma-algebras on a probability space $(\Omega,P)$ . If, for every $i\in\iota$ , the sigma-algebra generated by all strict predecessors is independent of $m_i$ ,

\left(\bigvee_{j<i}m_j\right)\perp m_i,

then the family $(m_i)_{i\in\iota}$ is mutually independent:

\operatorname{iIndep}(m_i)_{i\in\iota}\,P.

This is a general chain criterion for mutual independence and is used to organize Brownian increment independence.

Original Lean declaration header

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

theorem iIndep_of_indep_biSup_lt {ι : Type*} [LinearOrder ι] {m : ι → MeasurableSpace Ω}
    (h : ∀ i, Indep (⨆ j < i, m j) (m i) P) : iIndep m P

Module navigation

Module: LevyStochCalc/Brownian/MathlibBridge.

Next: hasLaw_eval.

Source excerpt, lines 43–44 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_MathlibBridge

/-
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/MathlibBridge.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} [MeasureTheory.IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.iIndep_of_indep_biSup_lt_readable {ι : Type*} [LinearOrder ι] {m : ι → MeasurableSpace Ω}
    (h : ∀ i, Indep (⨆ j < i, m j) (m i) P) : iIndep m P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/MathlibBridge.lean#L41-L63 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.MathlibBridge`, declaration `LevyStochCalc.Brownian.iIndep_of_indep_biSup_lt`; 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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