Splitting of Liftings in Product Spaces

Abstract

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 ${\mathfrak A}\otimes\mfB$. Moreover, let $\{({\mathfrak{A}}_y,S_y):y\in Y\}$ 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 $\mfC$ be a sub-$\sigma$-algebra of $\mathfrak{A}\cap {\bigcap }_{y\in Y}{\mathfrak{A}}_y$. For $f\in\mcL^{\infty}(R)$ I investigate the relationship between the $Y$-sections $[{\mathbb E}_{\mfC\otimes\mfB}(f)]^y$ of ${\mathbb E}_{\mfC\otimes\mfB}(f)$ (the conditional expectation of $f$ with respect to $\mfC\otimes\mfB$) and the conditional expectations of $f^y$ with respect $\mfC$ and $S_y$. Moreover I prove the existence of a lifting $\pi$ on $\mcL^{\infty}(\wh{R})$ ($\wh{R}$ is the completion of $R$) and liftings ${\sigma }_y$ on $\mcL^{\infty}(\wh{S_y})$, $y\in Y$, 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.