Geodesic
INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES
Forum of Mathematics, Sigma, Tome 4 (2016)

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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. 
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     author = {NATHANAEL ACKERMAN and CAMERON FREER and REHANA PATEL},
     title = {INVARIANT {MEASURES} {CONCENTRATED} {ON} {COUNTABLE} {STRUCTURES}},
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NATHANAEL ACKERMAN; CAMERON FREER; REHANA PATEL. INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.15

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