&lade in United States of America

Denote by S,, the set (x; 1 ~ji(x)~@> n), and put M, = m(S,). Then 2 M, converges. It is clear that the same holds if 1 $fk(x) 1 > n is replaced by I$ fx{x) I> c-n (replace fk(x) by c.fk(x)). It was conjectured that the conditions (1) and (2) are necessary for the convergence of 2 M, . Dr. Chung pointed it out to me that in this form the conjecture is ina”zurate; to see this it suffices to put fk(x) = :(l + rk(x)) where Q(Z) is the kth Rademacher function. Clearly If&r) 1 < 1; thus M, = 0, thus 2 M,, converges, but s 00 t dG(t) # 0. On the other hand we shall show n-l in the present note that the cozjecture of Hsu and Robbins is essentially correct. In fact we prove THEOREM I. The necessary and m&at condition for the convergence of 2 M, is that