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 [Positivity 19 (2015), pp. 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 [Positivity 14 (2010), pp. 859–875]. An $R(T)$ valued norm, for the cases of $p=1,\infty ,$ was introduced on these spaces in [J. Math. Anal. Appl. 456 (2017), pp. 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 [J. Math. Anal. Appl. 447 (2017), pp. 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 $\mathcal {L}^{\infty }(T)$.