On individual convergence in the space of $\tau$-measurable operators

Given a semifinite von Neumann algebra $(\mathcal M,\tau)$, we show that the spaces $L^0(\mathcal M,\tau)$ and $\mathcal R_\tau$ are complete with respect to the almost uniform and bilaterally almost uniform convergences in $L^0(\mathcal M,\tau)$ and discuss some applications of these results to the noncommutative ergodic theory.