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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces
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by Wen-Chi Kuo, David F. Rodda and Bruce A. Watson PDF
Proc. Amer. Math. Soc. 147 (2019), 1597-1603 Request permission

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 $\mathcal {L}^{\infty }(T)$.
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Additional Information
  • Wen-Chi Kuo
  • Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
  • MR Author ID: 744819
  • David F. Rodda
  • Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
  • Bruce A. Watson
  • Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
  • MR Author ID: 649582
  • ORCID: 0000-0003-2403-1752
  • Received by editor(s): February 22, 2018
  • Received by editor(s) in revised form: July 25, 2018, and July 26, 2018
  • Published electronically: December 12, 2018
  • Additional Notes: The first author was supported in part by National Research Foundation of South Africa grant no. CSUR160503163733.
    The second author was supported in part by National Research Foundation of South Africa grant no. 110943.
    The third author was supported in part by the Centre for Applicable Analysis and Number Theory and by National Research Foundation of South Africa grant IFR170214222646 with grant no. 109289.
  • Communicated by: Stephen Dilworth
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 1597-1603
  • MSC (2010): Primary 46B40, 60F15, 60F25
  • DOI: https://doi.org/10.1090/proc/14341
  • MathSciNet review: 3910424