Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces

Abstract

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 $\mathcal{L}^{\infty}(T)$.