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, 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.