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arXiv · 1206.4011

Invariant measures concentrated on countable structures

Nathanael Ackerman·Cameron Freer·Rehana Patel
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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.

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Nathanael Ackerman, Cameron Freer, Rehana Patel. 2016-06-28. Invariant measures concentrated on countable structures. https://doi.org/10.1017/fms.2016.15

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