On the causality and $K$-causality between measures

Drawing from our earlier works on the notion of causality for nonlocal phenomena, we propose and study the extension of the Sorkin–Woolgar relation $K^{+}$ onto the space of Borel probability measures on a given spacetime. We show that it retains its fundamental properties of transitivity and closedness. Furthermore, we list and prove several characterizations of this relation, including the `nonlocal’ analogue of the characterization of $K^{+}$ in terms of time functions. This generalizes and casts new light on our earlier results concerning the causal precedence relation $J^{+}$ between measures.