Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $Σ$ of $\mathcal{L}_{ω_1, ω}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection of models of $Σ$ with underlying set $\mathbb{N}$. Restricting to $\mathcal{L}_{ω, ω}(L)$, this answers an open question of Gaifman from 1964, via a translation between $S_\infty$-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.