Abstract

We prove that if (X,A,P) is an arbitrary probability space with countably generated σ-algebra A, (Y,B,Q) is an arbitrary complete probability space with a lifting ρ and ̂R is a complete probability measure on A ̂⊗R B determined by a regular conditional probability {Sy:y ∈ Y} on A with respect to B, then there exist a lifting π on (X × Y,A ̂⊗R B, ̂R) and liftings σy on (X,Ây, ̂Sy), y ∈ Y, such that, for every E ∈ A ̂⊗R B and every y ∈ Y, [π(E)] y = σy([π(E)] y). 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 ̟ on (X × Y,A ̂⊗R B, ̂R), ρ ′ on (Y,B, ̂Q) and σy on (X,Ây, ̂Sy), y ∈ Y, such that, for every E ∈ A ̂⊗R B and every y ∈ Y, [̟(E)] y = σy([̟(E)] y) and ̟(A × B) = y∈ρ ′ (B) σy(A) × {y} if A × B ∈ A × B. Both results are generalizations of Musia̷l, 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 ̂R-measurable stochastic processes into their ̂R-measurable modifications. 1. Preliminaries. If (Z,Z,S) is a probability space, then we denote by ̂Z the completion of Z with respect to S and by ̂ S the completion of S. We write L ∞ (S): = L ∞ (Z,Z,S) for the space of bounded Z-measurable realvalued functions. Functions equal a.e. are not identified. We use the notion of lower density and lifting in the sense of [7]. It is known (cf. [7]) that there is a 1–1 correspondence among liftings on Z with

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