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

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.