Countable infinitary theories admitting an invariant measure

TL;DR

This paper characterizes countable theories in infinitary logic that admit an invariant probability measure, answering a longstanding open question and extending previous results through new measure construction techniques.

Contribution

It provides a definable closure-based characterization of theories with invariant measures and introduces a method for constructing such measures from directed systems.

Findings

Characterization of theories with invariant measures using definable closure

Resolution of Gaifman's open question from 1964

Development of a machinery for building invariant measures from directed systems

Abstract

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection of models of $\Sigma$ 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 $S_\infty$-invariant measures and Gaifman's symmetric measure-models with strict equality. It also extends the known characterization in the case where $\Sigma$ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.