Countable infinitary theories admitting an invariant measure

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\mathrm{\Sigma }$ of $$\mathcal{L}${\omega_1, \omega}(L)$forwhichthereexistsan$S\infty$-invariant probability measure on the collection of models of $\mathrm{\Sigma }$ with underlying set $\mathbb{N}$. Restricting to ${\mathcal{L}}_{\omega ,\omega }(L)$, this answers an open question of Gaifman from 1964, via a translation between $S_{\mathrm{\infty }}$-invariant measures and Gaifman’s symmetric measure-models with strict equality. It also extends the known characterization in the case where $\mathrm{\Sigma }$ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.