  Let A be a standard Borel space, and consider the space A^{\bbN^{(k)}} of
A-valued arrays indexed by all size-k subsets of \bbN. This paper concerns
random measures on such a space whose laws are invariant under the natural
action of permutations of \bbN. The main result is a representation theorem for
such `exchangeable' random measures, obtained using the classical
representation theorems for exchangeable arrays due to de Finetti, Hoover,
Aldous and Kallenberg.
  After proving this representation, two applications of exchangeable random
measures are given. The first is a short new proof of the Dovbysh-Sudakov
Representation Theorem for exchangeable PSD matrices. The second is in the
formulation of a natural class of limit objects for dilute mean-field spin
glass models, retaining more information than just the limiting Gram-de Finetti
matrix used in the study of the Sherrington-Kirkpatrick model.
