Existence and Non-existence of a Finite Invariant Measure

Abstract

About fifty years ago, questions on the existence and non-existence of finite invariant measures were studied by various authors and from different directions. In this article, we examine these classical results and prove directly that all the conditions introduced by these authors are equivalent to each other. We begin at the fundamental level of a recurrent transformation whose properties can be strengthened to obtain the aforementioned classical results for the existence of a finite invariant measure. We conclude with the introduction of a new property, Strongly Weakly Wandering (sww) sequences, the existence of which is equivalent to the non-existence of a finite invariant measure. It is shown that every sww sequence is also an Exhaustive Weakly Wandering (eww) sequence for ergodic transformations. Although all ergodic transformations with no finite invariant measure are known to have eww sequences, there are exceedingly few actual examples for which explicit eww sequences can be exhibited. The significance of sww sequences is that it provides a condition which is easier to verify than the condition for eww sequences (Proposition 4.5). In a second paper, we will continue these studies and also connect them to some of the more recent derived conditions for finite invariant measures. The impetus for this work, began with the late Professor Shizuo Kakutani, with whom the authors worked and had many fruitful discussions on these topics.