Exchangeable random measures1

Let A be a standard Borel space, and consider the space Aℕ^(k) of A-valued arrays indexed by all size-k subsets of ℕ. This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations of ℕ. 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 positive semi-definite 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.