Representations of Aut(M)-Invariant Measures

In this paper we generalize the Aldous-Hoover-Kallenberg theorem concerning representations of distributions of exchangeable arrays via collections of measurable maps. We give criteria when such a representation theorem exists for arrays which need only be preserved by a closed subgroup of the symmetric group over $\mathbb{N}$. Specifically, for a countable structure M, with underlying set the $\mathbb{N}$, we introduce the notion of an "Aut(M)-recipe", which is an Aut(M)-invariant array obtained via a collection of measurable functions indexed by the Aut(M)-orbits in M. We further introduce the notion of a "free structure" and then show that if M is free then every Aut(M)-invariant measure on an Aut(M)-space is the distribution of an Aut(M)-recipe. We also show that if a measure is the distribution of an Aut(M)-recipe it must be the restriction of a measure on a free structure.