Time functions and $K$-causality between measures

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 $\mathcal{P}(\mathcal{M})$ 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 $\mathcal{P}(\mathcal{M})$ retains its property of antisymmetry for $\mathcal{M}$ stably causal.