Splitting of Liftings in Product Spaces II

Abstract

Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $\sigma$-algebra $\mfA\otimes\mfB$ and $\{(\mfA_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$. Then let $\mfA\dd\mfB$ be the $\sigma$-algebra generated $\mfA\otimes\mfB$ and by the family $\mcM:=\{E\subset{X\times{Y}}\colon \exists\;N\in\mfB_0\;\forall\;y\notin{N}\;\wh{S_y}(E^y)=0\}$ and $\wh{R_{\dd}}$ be the extension of $R$ such that $\mcM$ becomes the family of $\wh{R_*}$-zero sets ($\wh{S_y}$ is the completion of $S_y$ and $\mfB_0=\{B\in\mfB: Q(B)=0\}$). We prove that there exist a lifting $\pi$ on $\mcL^{\infty}(\wh{R_{\dd}})$ and liftings $\sigma_y$ on $\mcL^{\infty}(\wh{S_y})$ , $y\in Y$, such that \[ [\pi(f)]^y= \sigma_y\Bigl([\pi(f)]^y\Bigr) \qquad\mbox{for every} \quad y\in Y\quad\mbox{and every}\quad f\in\mcL^{\infty}(\wh{R_{\dd}}). \] In case of a separable $P$ and in case when $R\ll{P}\times{Q}$ a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a generalization and correction of \cite[Theorem 3.8]{mu25}.