Splitting of Conditional Expectations and Liftings in Product Spaces II

Let (X,A,P) and (Y,B,Q) be two probability spaces, R be their skew product on the product σ-algebra A⊗B and {(Ay,Sy):y∈Y} be a Q-disintegration of R. Then, let A⋇B be the σ-algebra generated A⊗B and by the family M:={E⊂X×Y:∃N∈B0∀y∉NSy^(Ey)=0} and R⋇ be the extension of R such that M becomes the family of R*-zero sets (Sy^ is the completion of Sy and B0={B∈B:Q(B)=0}). We prove that there exists a lifting π on L∞(R⋇) and liftings σy on L∞(Sy^), y∈Y, such that sections of π determined by Y are lifting invariant (in particular, the sections are measurable), i.e., [π(f)]y=σy[π(f)]y for every y∈Y and every f∈L∞(R⋇). In general, if π is an arbitrary lifting on the product, then some sections of π(f) may be even nonmeasurable. The main novelty of my paper lies in expanding the domain of the measure in the product to A⋇B and constructing on such a much larger abstract space the suitable lifting. Such expansions used to be made only in case of topological spaces, where product of marginal Borel sets was replaced by the Borel subsets of the product space. However, several topological technics are then applied, not approachable in the abstract case. The main theorem is a generalization of earlier lifting results, where either separability of A in the Frechet–Nikodým pseudometric was assumed or R≪P×Q. In case of a separable P and in the case when R≪P×Q, a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a strong generalization of earlier results (see the introduction) where it was proved only that the lifting modification of a measurable stochastic process (via the lifting constructed there) is again measurable.