Vector liftings for products of probability spaces and measurable modifications of stochastic processes

TL;DR

This paper studies vector liftings in probability spaces, establishing their existence for product spaces, and uses them to characterize stochastic processes with measurable modifications, improving measure extension techniques.

Contribution

It introduces the concept of vector liftings for all p, constructs strong product vector liftings, and characterizes stochastic processes with measurable modifications without relying on liftings.

Findings

Existence of strong product vector liftings for two-factor products.

Characterization of stochastic processes with measurable modifications.

Measure extension results applicable beyond complete measures.

Abstract

We investigate the properties of linear primitive liftings $\rho\colon \mathcal{L}^p(\mu)\to \mathcal{L}^p(\mu)$ for probability spaces $(X,\Sigma,\mu)$, which are linear maps selecting a representative from each class for almost everywhere equality. We call them vector liftings. They have the advantage over liftings or linear liftings that they exist for all $p\in[0,\infty]$, not only for $p=\infty$. Their relationship to products is still not clear, but we establish existence of (strong) product vector liftings for products of two factors. The vector liftings which are $2$-marginals with respect to a suitable product yielded a characterization of stochastic processes having a measurable modification modelled on one discovered by Musia{\l}, and led to a proof of the characterization that does not use liftings. The improvement relies on results on extending a measure by null sets that apply when the family of new null sets is an ideal of a $\sigma$-algebra larger than the domain of the measure, but not necessarily an ideal of the power set. These allow us to reduce or eliminate completeness assumptions on the measures from several of our results.