An effective version of Nadkarni's Theorem

Nadkarni’s Theorem asserts that for a countable Borel equivalence relation (CBER) exactly one of the following holds: (1) It has an invariant Borel probability measure or (2) it admits a Borel compression, i.e., a Borel injection that maps each equivalence class to a proper subset of it. We prove in this paper an effective version of Nadkarni’s Theorem, which shows that if a CBER is effectively Borel, then either alternative (1) above holds or else it admits an effectively Borel compression. As a consequence if a CBER is effectively Borel and admits a Borel compression, then it actually admits an effectively Borel compression. We also prove an effective version of the ergodic decomposition theorem. Finally a counterexample is given to show that alternative (1) above does not admit an effective version.