Abstract
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.
Citation
Clinton Conley. Benjamin Miller. "Measure reducibility of countable Borel equivalence relations." Ann. of Math. (2) 185 (2) 347 - 402, March 2017. https://doi.org/10.4007/annals.2017.185.2.1
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