“…The purpose of this paper is to prove that this is indeed the case: This also answers a question raised by Kechris in [9, p. 123, (IIIb) and (IVb)], who asked if unitary equivalence of probability measure preserving actions of Γ is Borel. We note that Theorem 1.1 stands in contrast to the recent result of Foreman et al [4], who have shown that the conjugacy relation on ergodic transformations in the group of measure preserving transformations Aut(X, μ) is complete analytic, and so cannot be Borel, when (X, μ) is a standard non-atomic probability space. The paper is organized as follows: First, in Section 2, we will recall various basic notions from the theory of von Neumann algebras and the direct integral theory of unitary representations, and establish a simple lemma that will facilitate hierarchy complexity calculations later.…”