  Let L_1 be the predual of a von Neumann algebra with a finite faithful normal
trace. We show that a bounded sequence in L_1 converges to 0 in measure if and
only if each of its subsequences admits another subsequence which converges to
0 in norm or spans $l^1$ "almost isometrically". Furthermore we give a
quantitative version of an essentially known result concerning the perturbation
of a sequence spanning $l^1$ isomorphically in the dual of a C$^*$-algebra.
