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

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