Orthogonal measures and ergodicity

Burgess-Mauldin have proven the Ramsey-theoretic result that continuous sequences ( μ c ) c ∈ 2 ℕ of pairwise orthogonal Borel probability measures admit continuous orthogonal subsequences. We establish an analogous result for sequences indexed by 2 N /E 0 , the next Borel cardinal. As a corollary,...

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Bibliographic Details
Published in:Israel journal of mathematics 2017-03, Vol.218 (1), p.83-99
Main Authors: Conley, Clinton T., Miller, Benjamin D.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Equivalence
Ergodic processes
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:Burgess-Mauldin have proven the Ramsey-theoretic result that continuous sequences ( μ c ) c ∈ 2 ℕ of pairwise orthogonal Borel probability measures admit continuous orthogonal subsequences. We establish an analogous result for sequences indexed by 2 N /E 0 , the next Borel cardinal. As a corollary, we obtain a strengthening of the Harrington-Kechris-Louveau E 0 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.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-017-1460-8