Countable Infinitary Theories Admitting an Invariant Measure

Let L be a countable language. We characterize, in terms of definable closure, those countable theories Σ of Lω1,ω(L) for which there exists an S∞-invariant probability measure on the collection of models of Σ with underlying set N. Restricting to Lω,ω(L), this answers an open question of Gaifman from 1964, via a translation between S∞-invariant measures and Gaifman’s symmetric measure-models with strict equality. It also extends the known characterization in the case where Σ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.