The Mean Ergodic Theorem in symmetric spaces
Volume 245 / 2019
Abstract
We investigate the validity of the Mean Ergodic Theorem in a symmetric Banach function space $E$ associated to an atomless Lebesgue probability space $(\Omega , \nu )$. We show that the Mean Ergodic Theorem holds if and only if $E$ is separable. That is, if $T:\Omega \to \Omega $ is a measure preserving bijection then the Cesàro averages of $\{ f \circ T^k \}_{k \ge 0}$ converge in a symmetric Banach function space $E$ for every $f \in E$ if and only if $E$ is separable. When $E$ is non-separable the Cesàro averages may converge in $E$ for some $f \in E$, but not all. It is also possible that every $f \in E$ can have an equimeasurable copy whose Cesàro averages do converge in $E$. We demonstrate this using sufficient conditions intimately connected with the theory of singular traces.
