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Title: Some properties of algebras of real-valued measurable functions (English)
Author: Estaji, Ali Akbar
Author: Mahmoudi Darghadam, Ahmad
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 59
Issue: 5
Year: 2023
Pages: 383-395
Summary lang: English
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Category: math
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Summary: Let $ M(X, \mathscr{A})$ ($M^{*}(X, \mathscr{A})$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X, \mathscr{A})$, let $M_{K}(X, \mathscr{A})$ be the family of all $f\in M(X, \mathscr{A})$ such that ${{\,\mathrm{coz}}}(f)$ is compact, and let $M_{\infty }(X, \mathscr{A})$ be all $f\in M(X, \mathscr{A})$ that $\lbrace x\in X: |f(x)|\ge \frac{1}{n}\rbrace $ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X, \mathscr{A})$, we show that $M^{*}(X, \mathscr{A})$ is a realcompact subring of $M(X, \mathscr{A})$, and also $M(X, \mathscr{A})$ is a realcompact if and only if $(X, \mathscr{A})$ is a compact measurable space. For every nonzero real Riesz map $\varphi : M(X, \mathscr{A})\rightarrow \mathbb{R}$, we prove that there is an element $x_0\in X$ such that $\varphi (f) =f(x_0)$ for every $f\in M(X, \mathscr{A})$ if $(X, \mathscr{A})$ is a compact measurable space. We confirm that $M_{\infty }(X, \mathscr{A})$ is equal to the intersection of all free maximal ideals of $M^{*}(X, \mathscr{A})$, and also $M_{K}(X, \mathscr{A})$ is equal to the intersection of all free ideals of $M(X, \mathscr{A})$ (or $M^{*}(X, \mathscr{A})$). We show that $M_{\infty }(X, \mathscr{A})$ and $M_{K}(X, \mathscr{A})$ do not have free ideal. (English)
Keyword: real measurable function
Keyword: lattice-ordered ring
Keyword: realcompact measurable space
Keyword: real Riesz map
Keyword: free ideal
MSC: 12J15
MSC: 28A20
MSC: 54C30
idZBL: Zbl 07790554
idMR: MR4641953
DOI: 10.5817/AM2023-5-383
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Date available: 2023-08-15T13:36:56Z
Last updated: 2024-02-13
Stable URL: http://hdl.handle.net/10338.dmlcz/151795
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