Boundedness and convergence of martingales in rearrangement-invariant function spaces

Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (f_n)_{n \geqq 0}f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices.