Pointwise convergence of ergodic averages in Orlicz spaces
Pointwise convergence of ergodic averages in Orlicz spaces
We construct a sequence ${a_n}$ such that for any aperiodic measure-preserving system $(X, \Sigma, m, T)$ the ergodic averages \begin{equation*} A_Nf(x) = \frac{1}{N} \sum_{n=1}^N f\bigl(T^{a_n}x\bigr) \end{equation*} converge a.e. for all $f$ in $L \log\log(L)$ but fail to have a finite limit for an $f \in L^1$. In fact, we show …