Splitting of liftings in products of probability spaces

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 R̂ 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, R̂) and liftings σy on (X, Ây, Ŝy), y ∈ Y , such that, for every E ∈ A ⊗̂R B and every y ∈ Y , [π(E)] y = σy([π(E)] ). 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, R̂), ρ ′ on (Y,B, Q̂) and σy on (X, Ây, Ŝy), y ∈ Y , such that, for every E ∈ A ⊗̂R B and every y ∈ Y , [̟(E)] = σy([̟(E)] ) and ̟(A×B) = ⋃