On the existence of vector measures with given marginals (Q1396789)

The author proves a type of Strassen's theorem for measures taking values in the positive cone of a Banach lattice. For example, in the topological context, she proves the following. Let \(X_i\), \(i=1,2\), be a Polish space, \({\mathcal B}(X_i)\) be the Borel \(\sigma \)-field of \(X_i\), and \(B^+\) be the positive cone of a Banach lattice \(B\) with the order continuous norm. Let \(\mu _i:{\mathcal B}(X_i)\to B^+,\) \(i=1,2\), be countably additive vector measures with \(\mu _1(X_1)=\mu _2(X_2)=\alpha \). Suppose \(C\) is a closed subset of \(X_1\times X_2\), and \(v\in B^+\) satisfies \(v\leq \alpha \). The following two statements are equivalent: (i) there exists a countably additive vector measure \(\mu :{\mathcal B}(X_1\times X_2)\to B^+\), such that \(\mu \circ \pi _i^{-1}=\mu _i\) and \(\mu (C)\geq v\) (here \(\pi _i:X_1\times X_2\to X_i\) stands for the projection onto \(X_i\)); (ii) for all closed sets \(C_i\in {\mathcal B}(X_i)\) with \(C\subset \pi _1^{-1}(C_1)\cup \pi _2^{-1}(C_2)\) it holds that \(\mu _1(C_1)+\mu _2(C_2)\geq v\).