Existence of measurable versions of stochastic processes

Let $(X,\text{mfA},P)$, $(Y,\text{mfB},Q)$ be two arbitrary probability spaces and $\mathrm{\P }:=\{(\text{mfA},P_y):y\in Y\}$ be a regular conditional probability on $\text{mfA}$ with respect to $Q$. Denote by $R$ the skew product of $P$ and $Q$ determined by $\{P_y:y\in Y\}$ on the product $\sigma$-algebra $\text{mfA}\otimes \text{mfB}$ and by $\text{wh}R$ its completion. I prove that a process $\{{\xi }_y:y\in Y\}$ possesses an equivalent $\text{wh}R$-measurable version if and only if it is measurable with respect to a certain particular $\sigma$-algebra, larger than $\text{mfA}\otimes \text{mfB}$ and uniquely determined by $\mathrm{\P }$. It is known that not every process possesses an equivalent measurable version (cf. \cite[§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 version of a measurable process (in case of $R=P\times Q$), provided $Y$ is endowed with a separable pseudometric. The theorem is a strong generalization of \cite[Theorem 6.1]{smm} and \cite[Theorem 5.1]{mms1} where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process.