Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \rightarrow \operatorname{Aut}(X)$ is the group homomorphism that defines the $G$-action. Under our assumption, the von Neumann algebra $L^{\infty}(X, \mu) \overline{\rtimes}_{\alpha} G$ is a factor. One can refer to [this post][1] for more details. Does anyone know references that discuss necessary conditions for the factor $L^{\infty}(X, \mu) \overline{\rtimes}_{\alpha} G$ to be hyperfinite?

[1]:https://mathoverflow.net/questions/346219