Generic dynamics and monotone complete c*-algebras

Let R be any ergodic, countable generic equivalence relation on a perfect Polish space X. It follows from the main theorem of §1 that, modulo a meagre subset of X, R may be identified with the relation of orbit equivalence ensuing from a canonical action of Z. Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III AW*-factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in §3, we show that the Dyer factor is isomorphic to the Takenouchi factor.