  Drawing from the theory of optimal transport we propose a rigorous notion of
a causal relation for Borel probability measures on a given spacetime. To
prepare the ground, we explore the borderland between causality, topology and
measure theory. We provide various characterisations of the proposed causal
relation, which turn out to be equivalent if the underlying spacetime has a
sufficiently robust causal structure. We also present the notion of the
'Lorentz-Wasserstein distance' and study its basic properties. Finally, we
discuss how various results on causality in quantum theory, aggregated around
Hegerfeldt's theorem, fit into our framework.
