Hyper-u-amenability and Hyperfiniteness of Treeable Equivalence Relations

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations and we show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. As corollaries of this result, we obtain that if a countable Borel equivalence relation is either: 1. measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a σ -compact Polish space, 2. treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, 3. treeable, amenable and Borel bounded, then it is hyperfinite.