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Abstract: Given an omega-categorical structure M, we consider the problem of classifying Aut(M)-invariant Keisler measures, i.e. finitely-additive probability measures on definable sets. By passing to a more general question about invariant random expansions and making use of exchangeability from probability theory, we resolve this problem in many cases. Our techniques, namely the random placement construction and a sort of multi-amalgamation, are very reminiscent of techniques in structural Ramsey theory. This is joint work with Colin Jahel and Paolo Marimon.