On the Existence of a Measure Invariant Under a Transformation

Abstract

Transformations which leave a measure function invariant are of importance in many branches of mathematics, notably in dynamics and in the theory of probability. It is therefore of some interest to investigate under what circumstances an invariant measure is possible. This may be considered part of the general problem of determining conditions under which there exists in a space a measure which is invariant under a group of transformations acting on the space. This problem has been solved only in special cases. Thus in case the space is a locally compact separable topological group, considered as acted upon by the group of right (or left) translations by its own elements, Haar's theorem' asserts the existence of an invariant measure. We consider the case of a general complete separable metric space acted on either by a group consisting of an automorphism and its powers, or by a one-parameter group of automorphisms. We shall set up a method which yields a finite invariant measure whenever such a measure is possible. One must, of course, specify what measure functions are to be admitted. In all cases they will be required to be completely additive, defined at least for all Borel sets, with real values > 0, and not identically zero. For the most part, we shall be concerned with finite measures, that is, measure functions in which the whole space has finite measure. Only in ?4 shall we consider infinite measures. A measure is said to be invariant under a transformation T if m(TA) is defined and equal to m(A) whenever m(A) is defined. A measure is said to be invariant under a group of transformations if it is invariant under every transformation of the group. In the case of a compact metric space, Kryloff and Bogoliouboff2 have shown that every automorphism (that is, one-to-one bicontinuous transformation), and every one-parameter continuous group of automorphisms, possesses a finite