Resumen de Splitting of liftings in products of probability spaces
N.D. Macheras, K. Musial, W. Strauss
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 {Sy: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 σy 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 {Ty: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 σy 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.
