Splitting of conditional expectations and 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 \({{\mathfrak {A}}}\otimes {{\mathfrak {B}}}\). 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 \({{\mathfrak {C}}}\) be a sub-\(\sigma \)-algebra of \({{\mathfrak {A}}}\cap \bigcap _{y\in {Y}}{{\mathfrak {A}}}_y\). We prove that if \(f\in {{\mathcal {L}}}^{\infty }(R)\) and \({{\mathbb {E}}}_{{{\mathfrak {C}}}\otimes {{\mathfrak {B}}}}(f)\) is the conditional expectation of f with respect to \({{\mathfrak {C}}}\otimes {{\mathfrak {B}}}\), then for Q-almost every \(y\in {Y}\) the y-section \([{{\mathbb {E}}}_{{{\mathfrak {C}}}\otimes {{\mathfrak {B}}}}(f)]^y\) is a version of the conditional expectation of \(f^y\) with respect \({{\mathfrak {C}}}\) and \(S_y\). Moreover there exist a lifting \(\pi \) on \({{\mathcal {L}}}^{\infty }(\widehat{R})\) (\(\widehat{R}\) is the completion of R) and liftings \(\sigma _y\) on \({{\mathcal {L}}}^{\infty }(\widehat{S_y})\), \(y\in Y\), such that

$$\begin{aligned}{}[\pi (f)]^y= \sigma _y\Bigl ([\pi (f)]^y\Bigr ) \qquad \hbox {for all} \quad y\in Y\quad \hbox {and}\quad f\in {{\mathcal {L}}}^{\infty }(\widehat{R}). \end{aligned}$$

Both results are generalizations of Strauss et al. (Ann Prob 32:2389–2408, 2004), where \({{\mathfrak {A}}}\) was assumed to be separable in the Frechet-Nikodým pseudometric and of Macheras et al. (J Math Anal Appl 335:213–224, 2007), where R was assumed to be absolutely continuous with respect to the product measure \(P\otimes {Q}\). Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.