  Employing the notion of a coupling between measures, drawn from the optimal
transport theory, we study the extension of the Sorkin-Woolgar causal relation
$K^+$ onto the space $\mathscr{P}(\mathcal{M})$ of Borel probability measures
on a given spacetime $\mathcal{M}$. We show that Minguzzi's characterization of
$K^+$ in terms of time functions possesses a "measure-theoretic"
generalization. Moreover, we prove that the relation $K^+$ extended onto
$\mathscr{P}(\mathcal{M})$ retains its property of antisymmetry for
$\mathcal{M}$ stably causal.
