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Decompositions and measures on countable Borel equivalence relations

Part of: Set theory Ergodic theory

Published online by Cambridge University Press:  04 December 2020

RUIYUAN CHEN
RUIYUAN CHEN*
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL61801, USA
*
e-mail: ruiyuan@illinois.edu
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Abstract

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

Keywords

cardinal algebrascountable Borel equivalence relationsergodic decomposition

MSC classification

Primary: 03E15: Descriptive set theory 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3671 - 3703
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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