Pointwise Convergence of Ergodic Averages in Orlicz Spaces

converge a.e. for all f in L log log(L) but fail to have a finite limit for an f ∈ L. In fact, we show that for each Orlicz space properly contained in L, 1 ≤ q < ∞, there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f ∈ L . This extends the work of K. Reinhold, who, building on the work of A. Bellow, constructed a sequence for which the averages AN f(x) converge a.e. for every f ∈ L , p > 1, but diverge for some f ∈ L. Our method, introduced by Bellow and extended by Reinhold and M. Wierdl, is perturbation.