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Orthogonal measures and ergodicity

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Abstract

Burgess-Mauldin have proven the Ramsey-theoretic result that continuous sequences \({\left( {{\mu _c}} \right)_{c \in {2^\mathbb{N}}}}\) of pairwise orthogonal Borel probability measures admit continuous orthogonal subsequences. We establish an analogous result for sequences indexed by 2N/E0, the next Borel cardinal. As a corollary, we obtain a strengthening of the Harrington-Kechris-Louveau E0 dichotomy for restrictions of measure equivalence. We then use this to characterize the family of countable Borel equivalence relations which are non-hyperfinite with respect to an ergodic Borel probability measure which is not strongly ergodic.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Clinton T. Conley

  2. Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090, Wien, Austria

    Benjamin D. Miller

Authors
  1. Clinton T. Conley
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  2. Benjamin D. Miller
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Corresponding author

Correspondence to Clinton T. Conley.

Additional information

The first author was supported in part by NSF Grant DMS-1500906.

The second author was supported in part by FWF Grant P28153 and SFB Grant 878.

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Conley, C.T., Miller, B.D. Orthogonal measures and ergodicity. Isr. J. Math. 218, 83–99 (2017). https://doi.org/10.1007/s11856-017-1460-8

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  • DOI: https://doi.org/10.1007/s11856-017-1460-8

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