On Invariant Measures

Given a measurable transformation on a measure space one can ask whether or not there is an equivalent measure that is invariant under the transformation. This problem is discussed very thoroughly in Halmos' Lectures on ergodic theory, pp. 81-90, 97. The first result along these lines is due to E. Hopf who obtained necessary and sufficient conditions for the existence of a finite invariant measure. The condition is that the whole space is "bounded," i.e. that the space is not a "copy" of a subset of strictly smaller measure. ("Copy" is defined below.) Recently Hajian and Kakutani (the paper is not yet published) showed that Hopf s condition is equivalent to the nonexistence of a set of nonzero measure having infinitely many disjoint images under the powers of the transformation. In [3] Halmos proved that there was a sigma-finite invariant measure if and only if the space was the union of a countable number of "bounded" sets. It was not known however whether or not every transformation had this property. Our example shows that there are transformations that admit no equivalent invariant measures.