  Consider the set of probability measures with given marginal distributions on
the product of two complete, separable metric spaces, seen as a correspondence
when the marginal distributions vary. In problems of optimal transport,
continuity of this correspondence from marginal to joint distributions is often
desired, in light of Berge's Maximum Theorem, to establish continuity of the
value function in the marginal distributions, as well as stability of the set
of optimal transport plans. Bergin (1999) established the continuity of this
correspondence, and in this note, we present a novel and considerably shorter
proof of this important result. We then examine an application to an assignment
game (transferable utility matching problem) with unknown type distributions.
