  A natural representation of random graphs is the random measure. The
collection of product random measures, their transformations, and non-negative
test functions forms a general representation of the collection of non-negative
weighted random graphs, directed or undirected, labeled or unlabeled, where (i)
the composition of the test function and transformation is a non-negative edge
weight function, (ii) the mean measures encode edge density/weight and vertex
degree density/weight, and (iii) the mean edge weight, when square-integrable,
encodes generalized spectral and Sobol representations. We develop a number of
properties of these random graphs, and we give simple examples of some of their
possible applications.
