Convergence of Weighted Averages of Martingales in Banach Function Spaces

Let fs fn.nG1 be a martingale and wn.nG1 a sequence of positive numbers such that Wns nks1wkª`. Kazamaki and Tsuchikura proved that f converges in L p 1-p-`.if and only if the weighted average sn f ..nG1 of f converges in L p, where sn f. are given by 1 n sn f.s Wn k s1wkfk , ns1, 2, . . . . We shall investigate the convergence of f and sn f. in general Banach function spaces X. Our main result can be applied to the case where X is a rearrangement-invariant space, or X is a weighted L p-space with a weight function satisfying the condition Ap introduced by Izumisawa and Kazamaki.