  Strong convergence and convergence in probability were generalized to the
setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc
Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces
with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015),
647-657}}] as $T$-strong convergence and convergence in $T$-conditional
probability, respectively. Generalized $L^{p}$ spaces for the cases of
$p=1,2,\infty$, were discussed in the setting of Riesz spaces as
$\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson},
{Discrete stochastic integration in Riesz spaces}, {\em Positivity}, {\bf 14}
{(2010), 859-875}}]. An $R(T)$ valued norm, for the cases of $p=1,\infty,$ was
introduced on these spaces in [{{\sc W. Kuo, M. Rogans, B.A. Watson}, {Mixing
processes in Riesz spaces}, {\em Journal of Mathematical Analysis and
Application}, {\bf 456} {(2017), 992-1004}}] where it was also shown that
$R(T)$ is a universally complete $f$-algebra and that these spaces are
$R(T)$-modules. In [{{\sc Y. Azouzi, M. Trabelsi}, {$L^p$-spaces with respect
to conditional expectation on Riesz spaces}, {\em Journal of Mathematical
Analysis and Application}, {\bf 447} {(2017), 798-816}}] functional calculus
was used to consider $\mathcal{L}^{p}(T)$ for $p\in (1,\infty)$. In this paper
we prove the strong sequential completeness of the space $\mathcal{L}^{1}(T)$,
the natural domain of the conditional expectation operator $T$, and the strong
completeness of $\mathcal{L}^{\infty}(T)$.
