We construct a sequence a n such that for any aperiodic measure-preserving system ( X , Σ, m , T ) the ergodic averages converge a.e. for all f in L log?log( L ) but fail to have a finite limit for an f ∈ L 1. In fact, 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 ∈ L 1. Our method, introduced by A. Bellow and extended by K. Reinhold and M. Wierdl, is perturbation.
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