We obtain a condition on the underlying family of a-algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation. = fgP dp -f g"dp= ¡ga dp-[ gTdp> ap(D) -[ gT dp. J J E\C J J E\C JE\C