Orthogonal measures and ergodicity |
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Authors: | Clinton T Conley Benjamin D Miller |
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Institution: | 1.Department of Mathematical Sciences,Carnegie Mellon University,Pittsburgh,USA;2.Kurt G?del Research Center for Mathematical Logic,Universit?t Wien,Wien,Austria |
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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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