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INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES
"... Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the ..."
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Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit