Let (X, 2F, v) be a finite measure space and let T:X-»X be a measurable transformation. In this paper we prove that the averages A n f(x) = (n + l)~'Y.osi^nf(T'x) converge a.e. for every / in L p (dv), \, if and only if there exists a measure y equivalent to v such that the averages apply uniformly L p (dv) into weak-L p (dy). As a corollary, we get that uniform boundedness of the averages in V(dv) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do

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... is, we first study measures v equivalent to a finite invariant measure fj. and we prove that sup ns0 A n (dv/ dfi)" 1/<p " u <oo a.e. is a necessary and sufficient condition for the averages to converge a.e. for every / in L p {dv).