In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we prove
THEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such that
Then there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.
The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani (, Theorem 6).
Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.