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Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let $f*=su{p}_{N}1/N{\sum }_{m=0}^{N-1}f\circ T^m$. In this paper we mainly investigate the question of whether (i) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f*>t)}fd\mu |dt<\infty$ and whether (ii) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f>t)}fd\mu |dt<\infty$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables $f\circ T^m$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.