  We consider the optimal mass transportation problem in $\RR^d$ with
measurably parameterized marginals, for general cost functions and under
conditions ensuring the existence of a unique optimal transport map. We prove a
joint measurability result for this map, with respect to the space variable and
to the parameter. The proof needs to establish the measurability of some
set-valued mappings, related to the support of the optimal transference plans,
which we use to perform a suitable discrete approximation procedure. A
motivation is the construction of a strong coupling between orthogonal
martingale measures. By this we mean that, given a martingale measure, we
construct in the same probability space a second one with specified covariance
measure. This is done by pushing forward one martingale measure through a
predictable version of the optimal transport map between the covariance
measures. This coupling allows us to obtain quantitative estimates in terms of
the Wasserstein distance between those covariance measures.
