Invariant measures concentrated on countable structures

TL;DR

This paper characterizes when countable structures admit invariant measures, linking it to trivial definable closure and strong amalgamation in Fraisse limits, thus identifying new classes of structures with such measures.

Contribution

It provides a complete characterization of countable structures admitting invariant measures based on definable closure and amalgamation properties.

Findings

Structures with trivial definable closure admit invariant measures.

Fraisse limits with strong amalgamation admit invariant measures.

Identifies classes of structures that do and do not admit invariant measures.

Abstract

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.