Volume functions in general relativity

It is shown that on every spacetime there is a finite Borel measure such that open sets have positive measure and the topological boundary of the chronological past/future of every point has measure zero. Using this measure volume, functions are defined. It is shown that they are semicontinuous, and the set of points at which they are discontinuous is a union of nullgeodesics. The following causality conditions are characterized in terms of their properties: chronological, distinguishing, strongly causal, causally continuous, globally hyperbolic.