Measure reducibility of countable Borel equivalence relations


We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.


Clinton T. Conley

Carnegie Mellon University, Pittsburgh, PA

Benjamin D. Miller

Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Vienna, Austria