Splitting of conditional expectations and liftings in product spaces

Let (X, 𝔄,P) and (Y, 𝔅,Q) be two probability spaces and R be their skew product on the product σ -algebra 𝔄⊗𝔅 . Moreover, let {(𝔄_y,S_y):y∈Y} be a Q -disintegration of R (if 𝔄_y=𝔄 for every y∈Y , then we have a regular conditional probability on 𝔄 with respect to Q ) and let ℭ be a sub- σ -algebra of 𝔄∩⋂ _y∈Y𝔄_y . We prove that if f∈ℒ^∞(R) and 𝔼_ℭ⊗𝔅(f) is the conditional expectation of f with respect to ℭ⊗𝔅 , then for Q -almost every y∈Y the y -section [𝔼_ℭ⊗𝔅(f)]^y is a version of the conditional expectation of f^y with respect ℭ and S_y . Moreover there exist a lifting π on ℒ^∞(R) ( R is the completion of R ) and liftings σ _y on ℒ^∞(S_y) , y∈ Y , such that [π (f)]^y= σ _y ([π (f)]^y ) for all y∈ Y and f∈ℒ^∞(R). Both results are generalizations of Strauss et al. (Ann Prob 32:2389–2408, 2004), where 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⊗Q . Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.