Two dimensional probabilities with a given conditional structure

A properly measurable set P subset of X x M-1(Y) (where X, Y are Polish spa ces and M-1(Y) is the space of Borel probability measures on Y) is consider ed. Given a probability distribution lambda is an element of M-1(X) the pap er treats the problem of the existence of X x Y-valued random vector (xi, e ta) for which L(xi) = lambda and L(eta/xi = x) is an element of P-x lambda- almost surely that possesses moreover some other properties such as "L(xi, eta) has the maximal possible support" of "L(eta/xi = x)'s are extremal mea sures in P-x's". The paper continues the research started in [7].