On quasi-invariant measures in uniquely ergodic systems

Abstract

Let X be a compact metrizable space, ~ (X) its G-algebra of Borel sets, and let T be a homeomorphism of X. We denote for a Borel measure # on X by T/2 the measure that is obtained by setting TI~(A)=I~(T -1 A), A e ~ . There exists always a Borel probability measure # such that T/2=# [10, p. 76]. If there exists only one such invariant measure then T is called uniquely ergodic. A measure # is called quasi-invariant for T if T# is equivalent to/2. In this paper we show that every uniquely ergodic homeomorphism of a compact metrizable space whose invariant measure is non-atomic possesses more than countably many non-atomic quasi-invariant ergodic measure classes. The problem of describing such measure classes has arisen in the theory of unitary representations of locally compact groups [12, p. 650]. In fact, we shall prove a more precise result. To state this result we need more terminology. Let (Y, ~ ) be a Borel structure and let ~ be an equivalence relation in Y. It can happen that the G-algebra ~ contains sets A i, i eN , such that