  It is known that, for a positive Dunford-Schwartz operator in a
noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a
noncommutative Orlicz space with order continuous norm, the corresponding
ergodic averages converge bilaterally almost uniformly. We show that these
averages converge bilaterally almost uniformly in each noncommutative symmetric
space $E$ such that $\mu_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where
$\mu_t(x)$ is a non-increasing rearrangement of $x$. In particular, these
averages converge bilaterally almost uniformly in all noncommutative symmetric
spaces with order continuous norm.
