The amenability–hyperfiniteness conjecture for countable Borel equivalence relations

Let $\mathcal{E}$ be a countable Borel equivalence relation on a standard Borel space. It is Borel amenable if there are non-negative Borel functions $p_n:\mathcal{E}\to\mathbb{R}$ satisfying the normalization and asymptotic invariance conditions described in the source. It is hyperfinite if it is the increasing union of finite Borel equivalence relations whose classes have uniformly bounded size.

Amenability–hyperfiniteness conjecture. For countable Borel equivalence relations, Borel amenability is equivalent to hyperfiniteness.

Hyperforminiteness implies Borel amenability, while the converse is a central open problem in the subject. The conjecture was posed by Weiss for amenable group actions and by Kechris in the general setting.