W. Strauss, N. D. Macheras, K. Musial
Abstract
We prove that if (X,,P) is an arbitrary probability space with countably generated σ-algebra , (Y,,Q) is an arbitrary complete probability space with a lifting ρand R is a complete probability measure on ⊗_R determined by a regular conditional probability S_y:y∈ Y on with respect to , then there exist a lifting πon (X× Y, ⊗_R , R) and liftings σ_y on (X, _y, S_y), y∈ Y, such that, for every E∈ ⊗_R and every y∈ Y, [π(E)]^y=σ_y([π(E)]^y). 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 on (X× Y, ⊗_R , R), ρ' on (Y,, Q) and σ_y on (X, _y, S_y), y∈ Y, such that, for every E∈ ⊗_R and every y∈ Y, [(E)]^y=σ_y([(E)]^y) and (A× B)=_y∈ρ'(B)σ_y(A)×y A× 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 R-measurable stochastic processes into their R-measurable modifications.