  In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and
Holden, a new class of sparse random graph processes based on the concept of
graphexes over $\sigma$-finite measure spaces has been introduced. In this
paper, we introduce a metric for graphexes that generalizes the cut metric for
the graphons of the dense theory of graph convergence. We show that a sequence
of graphexes converges in this metric if and only if the sequence of graph
processes generated by the graphexes converges in distribution. In the course
of the proof, we establish a regularity lemma and determine which sets of
graphexes are precompact under our metric. Finally, we establish an
identifiability theorem, characterizing when two graphexes are equivalent in
the sense that they lead to the same process of random graphs.
