Nonstandard Analysis of the Behavior of Ergodic Means of Dynamical Systems on Very Big Finite Probability Spaces

In this chapter we discuss the behavior of ergodic means of discrete time dynamical systems on a very big finite probability space Y (discrete dynamical systems below). The G. Birkhoff Ergodic Theorem states the eventual stabilization of ergodic means of integrable functions for almost all points of the probability space. The trivial proof of this theorem for the case of finite probability spaces shows that this stabilization happens for those time intervals, whose length n exceeds significantly the cardinality \(|Y|\) of Y, i.e., \(\frac{n}{|Y|}\) is a very big number. For the case of very big number \({|Y|}\) we introduce the class of S-integrable functions and we prove that the ergodic means of these functions exhibit a regular behavior even for intervals whose length is comparable with \({|Y|}\).