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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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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Horizontal Egorov property of Riesz spaces
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by Mikhail Popov PDF
Proc. Amer. Math. Soc. 149 (2021), 323-332 Request permission


We say that a Riesz space $E$ has the horizontal Egorov property if for every net $(f_\alpha )$ in $E$, order convergent to $f \in E$ with $|f_\alpha | + |f| \le e \in E^+$ for all $\alpha$, there exists a net $(e_\beta )$ of fragments of $e$ laterally convergent to $e$ such that for every $\beta$, the net $\bigl (|f - f_\alpha | \wedge e_\beta \bigr )_\alpha$ $e$-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space $E$ is said to satisfy the weak distributive law if for every $e \in E^+ \setminus \{0\}$ the Boolean algebra $\mathfrak {F}_e$ of fragments of $e$ satisfies the weak distributive law; that is, whenever $(\Pi _n)_{n \in \mathbb N}$ is a sequence of partitions of $\mathfrak {F}_e$, there is a partition $\Pi$ of $\mathfrak {F}_e$ such that every element of $\Pi$ is finitely covered by each of $\Pi _n$ (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net $(f_\alpha )$ order convergent to $f$ in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net $(v_\beta )$ and a net $(u_{\alpha , \beta })_{(\alpha , \beta ) \in A \times B}$, which uniformly tends to zero for every fixed $\beta$ such that $|f - f_\alpha | \le u_{\alpha , \beta } + v_\beta$ for all $\alpha , \beta$.
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Additional Information
  • Mikhail Popov
  • Affiliation: Institute of Exact and Technical Sciences, Pomeranian University in Słupsk, ul. Arciszewskiego 22d, PL-76-200 Słupsk, Poland; Vasyl Stefanyk Precarpathian National University, 57 Shevchenko str, Ivano-Frankivsk, 76018 Ukraine.
  • MR Author ID: 192683
  • ORCID: 0000-0002-3165-5822
  • Email:
  • Received by editor(s): March 14, 2020
  • Received by editor(s) in revised form: April 18, 2020, June 8, 2020, and June 11, 2020
  • Published electronically: October 20, 2020
  • Communicated by: Stephen Dilworth
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 323-332
  • MSC (2010): Primary 46A40; Secondary 46B42
  • DOI:
  • MathSciNet review: 4172608