Finding measures with given marginals

Let (A, X) and (B, Y) be measurable spaces and let V be a Dedekind sigma -complete vector lattice. Let mu (1) and mu (2) be measures defined on A and B, respectively, and taking their values in the positive cone of V. We define sigma -additivity of V-valued measures with respect to the order structure of V. Let A x B be the sigma -field generated by A and B. It is shown here that classical results of Strassen can be generalized to this situation. In particular, when mu (1)(X) = mu (2)(Y), there exists a V-valued sigma -additive measure mu on A x B such that mu (A x Y) = mu (1)(A) and mu (X x B) = mu (2)(B) if mu (1) is sigma -additive, mu (2) is sigma -compact and V satisfies the lattice condition of being weakly sigma -distributive. When V is Dedekind complete and satisfies the stronger property of weak (sigma, infinity)-distributivity then analogous results hold with mu (2) satisfying the weaker property of being completely compact.