  We show that for each Orlicz space properly contained in L^1 there is a
sequence along which the ergodic averages converge for functions in the Orlicz
space, but diverge for all f in L^1. This extends the work of K. Reinhold, who,
building on the work of A. Bellow,constructed a sequence for which the averages
converge a.e. for every f in L^p, p>q, but diverge for some f in L^q. Our
method, introduced by Bellow and extended by Reinhold and M. Wierdl, is
perturbation.
