Hyperfinite measure-preserving actions of countable groups and their model theory

We give a shorter proof of a theorem of G. Elek stating that two hyperfinite measure-preserving actions of a countable group on standard probability spaces are approximately conjugate if and only if they have the same invariant random subgroup. We then use this theorem to study model theory of hyperfinite measure-preserving actions of countable groups on probability spaces. This work generalizes the model-theoretic study of automorphisms of probability spaces conducted by I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov.