  It is well known that given two probability measures $\mu$ and $\nu$ on
$\mathbb{R}$ in convex order there exists a discrete-time martingale with these
marginals. Several solutions are known (for example from the literature on the
Skorokhod embedding problem in Brownian motion). But, if we add a requirement
that the martingale should minimise the expected value of some functional of
its starting and finishing positions then the problem becomes more difficult.
Beiglb\"{o}ck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow
measure which induces a family of martingale couplings, and solves the optimal
martingale transport problem for a class of bivariate objective functions. In
this article we extend their (existence and uniqueness) results by providing an
explicit construction of the shadow measure and, as an application, give a
simple proof of its associativity.
