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RECURRENCE AND THE EXISTENCE OF INVARIANT MEASURES

Part of: Set theory Classical measure theory

Published online by Cambridge University Press:  15 February 2021

MANUEL J. INSELMANN  and
BENJAMIN D. MILLER
MANUEL J. INSELMANN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF VIENNA OSKAR MORGERNSTERN PLATZ 1, 1090VIENNA, AUSTRIAE-mail: manuel.inselmann@univie.ac.atE-mail: benjamin.miller@univie.ac.atURL: https://homepage.univie.ac.at/benjamin.miller
BENJAMIN D. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF VIENNA OSKAR MORGERNSTERN PLATZ 1, 1090VIENNA, AUSTRIAE-mail: manuel.inselmann@univie.ac.atE-mail: benjamin.miller@univie.ac.atURL: https://homepage.univie.ac.at/benjamin.miller
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Abstract

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

Keywords

invariant measurerecurrencetransiencewandering

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 60 - 76
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Copyright
© The Association for Symbolic Logic 2021

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