Given a semifinite von Neumann algebra
M
\mathcal M
equipped with a faithful normal semifinite trace
τ
\tau
, we prove that the spaces
L
0
(
M
,
τ
)
L^0(\mathcal M,\tau )
and
R
τ
\mathcal R_\tau
are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in
L
0
(
M
,
τ
)
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
(
M
,
τ
)
L^1(\mathcal M,\tau )
can be extended to pointwise convergence of such nets in any fully symmetric space
E
⊂
R
τ
E\subset \mathcal R_\tau
, in particular, in any space
L
p
(
M
,
τ
)
L^p(\mathcal M,\tau )
,
1
≤
p
>
∞
1\leq p>\infty
. Some applications of these results in the noncommutative ergodic theory are discussed.