Splitting of Liftings in Product Spaces

Let $(X,\mathfrak{A},P)$ and $(Y,\mathfrak{B},Q)$ be two probability spaces and $R$ be their skew product on the product $\sigma$-algebra $\mathfrak{A}\otimes \text{mfB}$. Moreover, let $\{({\mathfrak{A}}_y,S_y)\text{\hspace{0.17em}⁣}:y\in Y\}$ be a $Q$-disintegration of $R$ (if ${\mathfrak{A}}_y=\mathfrak{A}$ for every $y\in Y$, then we have a regular conditional probability on $\mathfrak{A}$ with respect to $Q$) and let $\text{mfC}$ be a sub-$\sigma$-algebra of $\mathfrak{A}\cap {\bigcap }_{y\in Y}{\mathfrak{A}}_y$. For $f\in {\text{mcL}}^{\mathrm{\infty }}(R)$ I investigate the relationship between the $Y$-sections $[{\mathbb{E}}_{\text{mfC}\otimes \text{mfB}}(f){]}^y$ of ${\mathbb{E}}_{\text{mfC}\otimes \text{mfB}}(f)$ (the conditional expectation of $f$ with respect to $\text{mfC}\otimes \text{mfB}$) and the conditional expectations of $f^y$ with respect $\text{mfC}$ and $S_y$. Moreover I prove the existence of a lifting $\pi$ on ${\text{mcL}}^{\mathrm{\infty }}(\text{wh}R)$ ($\text{wh}R$ is the completion of $R$) and liftings ${\sigma }_y$ on ${\text{mcL}}^{\mathrm{\infty }}(\text{wh}{S}_y)$, $y\in Y$, such that \begin{equation*} [π(f)]^y= σ_y\Bigl([π(f)]^y\Bigr) \qquad\mbox{for all} \quad y\in Y\quad\mbox{and}\quad f\in\mcL^{\infty}(\wh{R}). \end{equation*} As an application a characterization of stochastic processes possessing an equivalent measurable version is presented.