  We consider Lorentzian manifolds as examples of partially ordered measure
spaces, sets endowed with compatible partial order relations and measures, in
this case given by the causal structure and the volume element defined by each
Lorentzian metric. This places the structure normally used to describe
spacetime in geometrical theories of gravity in a more general context, which
includes the locally finite partially ordered sets of the causal set approach
to quantum gravity. We then introduce a function characterizing the closeness
between any two partially ordered measure spaces and show that, when restricted
to compact spaces satisfying a simple separability condition, it is a distance.
In particular, this provides a quantitative, covariant way of describing how
close two manifolds with Lorentzian metrics are, or how manifoldlike a causal
set is.
