Splitting of liftings in products of probability spaces

Abstract

We prove that if $(X,{\mathfrak{A}},P)$ is an arbitrary probability space with countably generated σ-algebra ${\mathfrak{A}}$, $(Y,{\mathfrak{B}},Q)$ is an arbitrary complete probability space with a lifting ρ and $\widehat {R}$ is a complete probability measure on ${\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ determined by a regular conditional probability $\{S_y:y∈Y\}$ on ${\mathfrak{A}}$ with respect to ${\mathfrak{B}}$, then there exist a lifting π on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$ and liftings $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\pi(E)]^{y}=\sigma_{y}\bigl([\pi(E)]^{y}\bigr).$$ Assuming the absolute continuity of $R$ with respect to $P⊗Q$, we prove the existence of a regular conditional probability $\{T_y:y∈Y\}$ and liftings ϖ on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$, ρ' on $(Y,\mathfrak{B},\widehat {Q})$ and $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\varpi(E)]^{y}=\sigma_{y}\bigl([\varpi(E)]^{y}\bigr)$$ and $$\varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_{y}(A)\times\{y\}\qquad\mbox{if }A\times B\in{\mathfrak{A}}\times{\mathfrak{B}}.$$ Both results are generalizations of Musiał, Strauss and Macheras [Fund. Math. 166 (2000) 281–303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert $\widehat {R}$-measurable stochastic processes into their $\widehat {R}$-measurable modifications.