Invariant measures concentrated on countable structures
Abstract
Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2012
 arXiv:
 arXiv:1206.4011
 Bibcode:
 2012arXiv1206.4011A
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics;
 Mathematics  Probability;
 03C98;
 60G09;
 37L40;
 05C80;
 03C75;
 62E10;
 05C63
 EPrint:
 46 pages, 2 figures. Small changes following referee suggestions