In this section, we study the construction of probability measures with given marginals. In particular, this will yield the missing implication in the characterization of uniform preference in Theorem 2.57, but the results in this section are of independent interest. We focus on the following basic question: Suppose μ1 and μ2 are two probability measures on S, and Λ is a convex set of probability measures on S × S; when does Λ contain some μ which has μ1 and μ2 as marginals?

The answer to this question will be given in a general topological setting. Let S be a Polish space, and let us fix a continuous function ψ on S with values in [1, ∞). As in Section 2.2 and in Appendix A.6, we use ψ as a gauge ...

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