Ditzen’s effective version of Nadkarni’s theorem

Abstract

Nadkarni’s theorem asserts that for a countable Borel equivalence relation (CBER), exactly one of the following holds: (1) It has an invariant probability Borel measure or (2) it admits a Borel compression, i.e., a Borel injection that maps each equivalence class to a proper subset of it. An effective version of Nadkarni’s theorem was included in Ditzen’s unpublished PhD thesis, where it is shown that if a CBER is effectively Borel, then either alternative (1) above holds or else it admits an effectively Borel compression. In his thesis, Ditzen also proves an effective version of the ergodic decomposition theorem. These notes contain an exposition of these results. We include Ditzen’s proof of the effective Nadkarni theorem, and use this construction to provide a different proof of the effective ergodic decomposition theorem. In addition, we construct a counterexample to show that alternative (1) above does not admit an effective version.