Independence Preserved By Almost-Everywhere Convergence

Proved

LevyStochCalc.Probability.indep_comap_of_tendsto_ae_readable

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

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

Let $m\leq m_\Omega$ be sigma-algebras on a probability space $(\Omega,P)$ , and let $\mathcal X$ be a topological Borel space with closed-set outer approximation. Suppose $X_n:\Omega\to\mathcal X$ are measurable, $Y:\Omega\to\mathcal X$ is measurable, and $m$ is independent of the comap sigma-algebra of every $X_n$ . If

X_n(\omega)\longrightarrow Y(\omega)

for $P$ -almost every $\omega$ , then $m$ is independent of the comap sigma-algebra generated by $Y$ . The result transfers independence through an almost-everywhere pointwise limit and is useful for limiting constructions of random variables.

Original Lean declaration header

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

theorem indep_comap_of_tendsto_ae (hm : m ≤ mΩ) {X : ℕ → Ω → 𝓧}
    {Y : Ω → 𝓧} (hX : ∀ n, Measurable[mΩ] (X n)) (hY : Measurable[mΩ] Y)
    (hind : ∀ n, Indep m (MeasurableSpace.comap (X n) inferInstance) P)
    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
    Indep m (MeasurableSpace.comap Y inferInstance) P

Module navigation

Module: LevyStochCalc/Probability/IndepLimit.

Supporting theorems: setIntegral_eq_measure_mul_of_indep.

Previous: setIntegral_eq_measure_mul_of_indep.

Source excerpt, lines 56–60 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Probability_IndepLimit

/-
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/Probability/IndepLimit.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory Filter Topology
variable {Ω 𝓧 : Type*} {m mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  [TopologicalSpace 𝓧] [MeasurableSpace 𝓧] [BorelSpace 𝓧] [HasOuterApproxClosed 𝓧]

Formal statement


theorem LevyStochCalc.Probability.indep_comap_of_tendsto_ae_readable (hm : m ≤ mΩ) {X : ℕ → Ω → 𝓧}
    {Y : Ω → 𝓧} (hX : ∀ n, Measurable[mΩ] (X n)) (hY : Measurable[mΩ] Y)
    (hind : ∀ n, Indep m (MeasurableSpace.comap (X n) inferInstance) P)
    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
    Indep m (MeasurableSpace.comap Y inferInstance) P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Probability/IndepLimit.lean#L54-L97 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Probability.IndepLimit`, declaration `LevyStochCalc.Probability.indep_comap_of_tendsto_ae`; 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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