Vector liftings for products of probability spaces and measurable modifications of stochastic processes
Maxim R. Burke, Nikolaos D. Macheras, Werner Strauss
TL;DR
The paper develops a comprehensive framework for vector liftings across product spaces, using extensions by null sets to construct product liftings and analyze their marginals. It proves key existence results for product strong vector liftings, and shows how 2-marginals underpin measurable modifications of stochastic processes, often relaxing completeness assumptions. It also introduces a robust Fubini-type apparatus for composing liftings via δ⊗η and investigates when such compositions preserve linear, multiplicative, or positive structure, tying together marginal theory, stochastic process modification, and tensor-like decompositions in L^p spaces.
Abstract
We investigate the properties of linear primitive liftings $ρ\colon \mathcal{L}^p(μ)\to \mathcal{L}^p(μ)$ for probability spaces $(X,Σ,μ)$, 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 Musiał, 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 $σ$-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.