Published online by Cambridge University Press: 28 June 2016
Let $L$ be a countable language. We say that a countable infinite
$L$ -structure
${\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of
$L$ -structures with the same underlying set as
${\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of
${\mathcal{M}}$ . We show that
${\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in
$\text{Aut}({\mathcal{M}})$ of an arbitrary finite tuple of
${\mathcal{M}}$ fixes no additional points. When
${\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of
${\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.