  Given a semifinite von Neumann algebra $\mathcal M$ equipped with a faithful
normal semifinite trace $\tau$, we prove that the spaces $L^0(\mathcal M,\tau)$
and $\mathcal R_\tau$ are complete with respect to pointwise, almost uniform
and bilaterally almost uniform, convergences in $L^0(\mathcal M,\tau)$. Then we
show that the pointwise Cauchy property for a special class of nets of linear
operators in the space $L^1(\mathcal M,\tau)$ can be extended to pointwise
convergence of such nets in any fully symmetric space $E\subset\mathcal
R_\tau$, in particular, in any space $L^p(\mathcal M,\tau)$, $1\leq p<\infty$.
Some applications of these results in the noncommutative ergodic theory are
discussed.
