Existence of measurable versions of stochastic processes

Abstract

Let $(X, \mfA,P)$, $(Y, \mfB,Q)$ be two arbitrary probability spaces and $\P:=\{(\mfA,P_y):y\in{Y}\}$ be a regular conditional probability (rcp) on $\mfA$ with respect to $Q$. Denote by $R$ the skew product of $P$ and $Q$ determined by $\P$ on the product $\sigma$-algebra $\mfA\otimes\mfB$ and by $\wh{R}$ its completion. I prove that if $(X, \mfA,P)$ is separable in the Fr\'{e}chet-Nikod\'{y}m pseudo-metric, then the stochastic process $\{\xi_y:y\in{Y}\}$ has an equivalent measurable modification if and only if it is measurable with respect to a certain particular $\sigma$-algebra larger than $\mfA\otimes\mfB$. The theorem is a strong generalization of \cite[Theorem 5.5]{mms2} and \cite[Theorem 6.1]{smm},where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process. It is known that not every process possesses an equivalent measurable modification (cf. \cite[Section 19.5]{St}). My approach is essentially different from earlier trials. It reverts to \cite[Theorem 3]{ta1}, where Talagrand proved existence of an equivalent separable modification of a measurable process (in case of $R=P\times{Q}$), provided $Y$ is endowed with a separable pseudometric.