Splitting of Liftings in Product Spaces II

Let $(X,\text{mfA},P)$ and $(Y,\text{mfB},Q)$ be two probability spaces, $R$ be their skew product on the product $\sigma$-algebra $\text{mfA}\otimes \text{mfB}$ and $\{({\text{mfA}}_y,S_y)\text{\hspace{0.17em}⁣}:y\in Y\}$ be a $Q$-disintegration of $R$. Then let $\text{mfA}\text{dd}\text{mfB}$ be the $\sigma$-algebra generated $\text{mfA}\otimes \text{mfB}$ and by the family $\text{mcM}:=\{E\subset X\times Y\text{\hspace{0.17em}⁣}:\mathrm{\exists }N\in {\text{mfB}}_0\mathrm{\forall }y\mathrm{\notin }N\text{wh}{S}_y(E^y)=0\}$ and $\text{wh}{R}_{\text{dd}}$ be the extension of $R$ such that $\text{mcM}$ becomes the family of $\text{wh}{R}_{\ast }$-zero sets ($\text{wh}{S}_y$ is the completion of $S_y$ and ${\text{mfB}}_0=\{B\in \text{mfB}:Q(B)=0\}$). We prove that there exist a lifting $\pi$ on ${\text{mcL}}^{\mathrm{\infty }}(\text{wh}{R}_{\text{dd}})$ and liftings ${\sigma }_y$ on ${\text{mcL}}^{\mathrm{\infty }}(\text{wh}{S}_y)$ , $y\in Y$, such that $$[\pi (f){]}^y={\sigma }_y([\pi (f){]}^y)\text{qquadfor}everyy\in Y\text{quadand}everyf\in {\text{mcL}}^{\mathrm{\infty }}(\text{wh}{R}_{\text{dd}})\mathrm{.}$$ 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}.