2018/03/22 by Wen-Chi Kuo, David F. Rodda, Kuo, Wen-Chi +4 · 1 citation
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA
paper · pdf · doi:10.48550/arxiv.1803.08538
arxiv created 2018/03/22 · openalex publication_date 2018/03/22 · arxiv updated 2018/03/26 · openalex created_date 2018/04/06 · openalex updated_date 2026/07/28
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 Lp spaces for the cases of p=1,2,∞, were discussed in the setting of Riesz spaces as Lp(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,∞, 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, Lp-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 Lp(T) for p∈ (1,∞). In this paper we prove the strong sequential completeness of the space L1(T), the natural domain of the conditional expectation operator T, and the strong completeness of L∞(T).