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Horizontal Egorov property of Riesz spaces


Author: Mikhail Popov
Journal: Proc. Amer. Math. Soc. 149 (2021), 323-332
MSC (2010): Primary 46A40; Secondary 46B42
DOI: https://doi.org/10.1090/proc/15235
Published electronically: October 20, 2020
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Abstract: 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 $ \vert f_\alpha \vert + \vert f\vert \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 (\vert f - f_\alpha \vert \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 $ \vert f - f_\alpha \vert \le u_{\alpha , \beta } + v_\beta $ for all $ \alpha , \beta $.


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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.
Email: misham.popov@gmail.com

DOI: https://doi.org/10.1090/proc/15235
Keywords: Riesz space, order convergence, uniform order convergence
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
Article copyright: © Copyright 2020 American Mathematical Society