A Generalized Shannon-McMillan Theorem for the Action of an Amenable Group on a Probability Space

Abstract

A generalization of the Shannon-McMillan Theorem ($L^1$ version) is obtained for the action of an amenable group on a probability space, thereby settling a conjecture of Pickel and Stepin. Interesting properties of the limit function are derived. The entropy of an action of an amenable group is defined.