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INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES

Part of: Stochastic processes Model theory Distribution theory Infinite-dimensional dissipative dynamical systems Graph theory

Published online by Cambridge University Press:  28 June 2016

NATHANAEL ACKERMAN ,
CAMERON FREER  and
REHANA PATEL
NATHANAEL ACKERMAN
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA; nate@math.harvard.edu
CAMERON FREER
Affiliation:
Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar Street, Cambridge, MA 02139, USA; freer@mit.edu
REHANA PATEL
Affiliation:
Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, MA 02492, USA; rehana.patel@olin.edu

Abstract

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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.

MSC classification

Primary: 03C98: Applications of model theory
Secondary: 60G09: Exchangeability 37L40: Invariant measures 05C80: Random graphs 03C75: Other infinitary logic 62E10: Characterization and structure theory 05C63: Infinite graphs
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e17
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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