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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Borel actions of Polish groups

Authors: Howard Becker and Alexander S. Kechris
Journal: Bull. Amer. Math. Soc. 28 (1993), 334-341
MSC: Primary 03E15; Secondary 03C75, 28E99, 54E99, 54H11
MathSciNet review: 1185149
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Abstract: We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analog of Tarski's theorem on paradoxical decompositions by showing that the existence of an invariant Borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many Borel pieces. We show that various natural versions of the Topological Vaught Conjecture are equivalent with each other and, in the case of the group of permutations of $ {\mathbb{N}}$, with the model-theoretic Vaught Conjecture for infinitary logic; this depends on our identification of the universal action for that group.

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Article copyright: © Copyright 1993 American Mathematical Society