Unique ergodicity for quasi-invariant measures

Abstract

Let (X, 5 p, #) be a standard measure space, and let T be a non-singular ergodic automorphism of (X, 5 P, #). If # is a T-invariant probability measure, there exists a T-invariant Borel set B c X with #(B)= 1, on which T is uniquely ergodic: this means that every T-invariant ergodic probabili ty measure v4:# on (X, 5 p) satisfies v(B)=0 (cf. [4] and [9]). If one leaves the setting of T-invariant probabili ty measures the situation is quite different. Suppose that the measure # is non-atomic, quasi-invariant and ergodic under T. Then one can show that for every T-invariant Borel set B c X with #(X\B)= 0 there exists a non-atomic o-finite measure v on (X, Y) which is quasi-invariant and ergodic under T, singular with respect to #, and which satisfies v(X\B)=O. In fact, one can even choose v to be T-invariant [7,9], or to satisfy certain other conditions [5]. In other words, quasi-invariant (even infinite invariant) measures cannot be uniquely ergodic. This problem has received attention at various levels of generality in [1-3, 5, 6]. The aim of this short note is to show that the concept of unique ergodicity nevertheless survives in a natural sense even in the case of quasiinvariant measures. In order to make this clear we need a definition (cf. [8]):