An 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 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.
Cite
@article{arxiv.2305.14518,
title = {An effective version of Nadkarni's Theorem},
author = {Alexander S. Kechris and Michael S. Wolman},
journal= {arXiv preprint arXiv:2305.14518},
year = {2023}
}
Comments
We have recently found out that the main results of the paper were included in the unpublished Caltech PhD thesis of Achim Ditzen, Definable Equivalence Relations on Polish Spaces (1992)