Countable abelian group actions and hyperfinite equivalence relations

An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations where all -equivalence classs are finite. In this article we establish the following theorem: if a countable abelian group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of .