Invariant measures concentrated on countable structures
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2012
- DOI:
- 10.48550/arXiv.1206.4011
- arXiv:
- arXiv:1206.4011
- Bibcode:
- 2012arXiv1206.4011A
- Keywords:
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- Mathematics - Logic;
- Mathematics - Combinatorics;
- Mathematics - Probability;
- 03C98;
- 60G09;
- 37L40;
- 05C80;
- 03C75;
- 62E10;
- 05C63
- E-Print:
- 46 pages, 2 figures. Small changes following referee suggestions