The Mean Ergodic Theorem in symmetric spaces

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 -> Omega is a measure preserving bijection then the Cesaro averages of {f o T-k}(k >= 0) converge in a symmetric Banach function space E for every f is an element of E if and only if E is separable. When E is non-separable the Cesaro averages may converge in E for some f is an element of E, but not all. It is also possible that every f is an element of E can have an equimeasurable copy whose Cesaro averages do converge in E. We demonstrate this using sufficient conditions intimately connected with the theory of singular traces.