Every equivalence relation with countable classes is the orbit relation of one permutation

Let $r$ be an equivalence relation on a set $X$ whose classes are countable. Then there is a bijection $T$ of $X$ such that $x \mathrel r y$ if and only if $y = T^n(x)$ for some $n \in \mathbb Z$: arrange each class as a finite cycle or as a copy of $\mathbb Z$.

Lodha and Moore define (p. 4): "Let $X$ be a Polish space and let $E \subseteq X^2$ be an equivalence relation which is Borel and which has countable equivalence classes. $E$ is $\mu$-amenable if, after discarding a $\mu$-measure $0$ set, $E$ is the orbit equivalence relation of an action of $\mathbb Z$." The definition does not say that the action of $\mathbb Z$ is Borel. Read without that, this theorem makes every such relation $\mu$-amenable, and Theorem 2.2 would be false; Theorem 2.2 says the orbit relation of a countable dense subgroup of $\mathrm{PSL}_2(\mathbb R)$ on the projective line is not amenable with respect to Lebesgue measure. The Lodha–Moore mission's definition (LodhaMoore.IsMuAmenable) takes the action to be by a Borel automorphism.