Marginals, measurable modifications of stochastic processes, and the product lifting problem

TL;DR

This paper investigates the problem of measurable lifting modifications for stochastic processes and explores the existence of marginals in product probability spaces, addressing the product lifting problem in a general setting.

Contribution

It introduces a general framework for the measurable lifting problem and reduces solutions to the existence of specific marginals in product spaces.

Findings

Reduction of the lifting problem to marginal existence

Analysis of product probability spaces and their marginals

Discussion on strong marginals in the context of stochastic processes

Abstract

This paper deals with the problem of measurable lifting modification for stochastic processes in its most general form and with the 'product lifting problem'. Solutions to the positive are reduced to the existence of marginals with respect to product probability spaces between the ordinary product and the product whose probability measure is the restriction of the skew product of the factor probabilities to the $\sigma$-algebra obtained by adjoing either the right or left nil-null sets to the ordinary product algebra. We discuss the problem of the existence of (strong) marginals.