Let T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.
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