On the continuity of correspondences on sets of measures with restricted marginals

Consider the set of probability measures on a product space with the proper ty that all have the same marginal distributions on the coordinate spaces. This set may be viewed as a correspondence, when the marginal distributions are varied. Here, it is shown that this correspondence is continuous. Nume rous problems in economics involve optimization over a space of measures wh ere one or more marginal distributions is given. Thus, for this class of pr oblem, Berge's theorem of the maximum is applicable: the set of optimizers is upper-hemicontinuous and the value of the optimal solution varies with t he parameters (marginals) continuously.