Means on equivalence relations

@article{Kechris2008MeansOE,
  title={Means on equivalence relations},
  author={Alexander S. Kechris and Benjamin D. Miller},
  journal={Israel Journal of Mathematics},
  year={2008},
  volume={163},
  pages={241-262}
}
Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.