Ergodic Theorem Without Invariant Measure

Abstract

The result presented in this paper is generalization of the celebrated ergodic theorem of G. D. Birkhoff.' Birkhoff's theorem, von Neumann's mean ergodic theorem, and other known results of ergodic theory deal with measure-preserving flows and describe their statistical properties. The condition of measure invrariance seems to play a very essential role in the entire theory, and the question arises whether it is possible to make any general statements about statistical behaviour of flows without assuming the invariance of the underlying measure. We shall answer this question in the affirmative by introducing an averaging process that can be applied to any flow which is free from dissipative parts. In the special case of a measure-preserving flow, our result coincides with Birkhoff's theorem.