On non-singular transformations of a measure space. I

Abstract

We consider a Lebesgue measure space (M, ∇, m). By an automorphism of (M, ∇, m) we mean a bi-measurable transformation of (M, ∇, m) that together with its inverse is non-singular with respeot to m. We study an equivalence relation between these automorphisms that we call the weak equivalence. Two automorphisms S and T are weakly equivalent if there is an automorphism U such that for almost all xε M U maps the S-orbit of x onto the T-orbit of U x. Ergodicity, the existence of a finite invariant measure, the existenoe of a σ-finite infinite invariant measure, and the non-existence of such measures are invariants of weak equivalenoe. In this paper and in its sequel we solve the problem of weak equivalenoe for a class of automorphisms that comprises all ergodic automorphisms that admit a σ-finite invariant measure, and also certain ergodic automorphisms that do not admit such a measure.