Some properties of algebras of real-valued measurable functions

Estaji, Ali Akbar, and Mahmoudi Darghadam, Ahmad. "Some properties of algebras of real-valued measurable functions." Archivum Mathematicum 059.5 (2023): 383-395. <http://eudml.org/doc/299132>.

@article{Estaji2023,
abstract = {Let $ M(X, \mathcal \{A\})$ ($M^\{*\}(X, \mathcal \{A\})$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X, \mathcal \{A\})$, let $M_\{K\}(X, \mathcal \{A\})$ be the family of all $f\in M(X, \mathcal \{A\})$ such that $\{\{\,\mathrm \{coz\}\}\}(f)$ is compact, and let $M_\{\infty \}(X, \mathcal \{A\})$ be all $f\in M(X, \mathcal \{A\})$ 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 $M(X, \mathcal \{A\})$, we show that $M^\{*\}(X, \mathcal \{A\})$ is a realcompact subring of $M(X, \mathcal \{A\})$, and also $M(X, \mathcal \{A\})$ is a realcompact if and only if $(X, \mathcal \{A\})$ is a compact measurable space. For every nonzero real Riesz map $\varphi : M(X, \mathcal \{A\})\rightarrow \mathbb \{R\}$, we prove that there is an element $x_0\in X$ such that $\varphi (f) =f(x_0)$ for every $f\in M(X, \mathcal \{A\})$ if $(X, \mathcal \{A\})$ is a compact measurable space. We confirm that $M_\{\infty \}(X, \mathcal \{A\})$ is equal to the intersection of all free maximal ideals of $M^\{*\}(X, \mathcal \{A\})$, and also $M_\{K\}(X, \mathcal \{A\})$ is equal to the intersection of all free ideals of $M(X, \mathcal \{A\})$ (or $M^\{*\}(X, \mathcal \{A\})$). We show that $M_\{\infty \}(X, \mathcal \{A\})$ and $M_\{K\}(X, \mathcal \{A\})$ do not have free ideal.},
author = {Estaji, Ali Akbar, Mahmoudi Darghadam, Ahmad},
journal = {Archivum Mathematicum},
keywords = {real measurable function; lattice-ordered ring; realcompact measurable space; real Riesz map; free ideal},
language = {eng},
number = {5},
pages = {383-395},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Some properties of algebras of real-valued measurable functions},
url = {http://eudml.org/doc/299132},
volume = {059},
year = {2023},
}

TY - JOUR
AU - Estaji, Ali Akbar
AU - Mahmoudi Darghadam, Ahmad
TI - Some properties of algebras of real-valued measurable functions
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 059
IS - 5
SP - 383
EP - 395
AB - Let $ M(X, \mathcal {A})$ ($M^{*}(X, \mathcal {A})$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X, \mathcal {A})$, let $M_{K}(X, \mathcal {A})$ be the family of all $f\in M(X, \mathcal {A})$ such that ${{\,\mathrm {coz}}}(f)$ is compact, and let $M_{\infty }(X, \mathcal {A})$ be all $f\in M(X, \mathcal {A})$ 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 $M(X, \mathcal {A})$, we show that $M^{*}(X, \mathcal {A})$ is a realcompact subring of $M(X, \mathcal {A})$, and also $M(X, \mathcal {A})$ is a realcompact if and only if $(X, \mathcal {A})$ is a compact measurable space. For every nonzero real Riesz map $\varphi : M(X, \mathcal {A})\rightarrow \mathbb {R}$, we prove that there is an element $x_0\in X$ such that $\varphi (f) =f(x_0)$ for every $f\in M(X, \mathcal {A})$ if $(X, \mathcal {A})$ is a compact measurable space. We confirm that $M_{\infty }(X, \mathcal {A})$ is equal to the intersection of all free maximal ideals of $M^{*}(X, \mathcal {A})$, and also $M_{K}(X, \mathcal {A})$ is equal to the intersection of all free ideals of $M(X, \mathcal {A})$ (or $M^{*}(X, \mathcal {A})$). We show that $M_{\infty }(X, \mathcal {A})$ and $M_{K}(X, \mathcal {A})$ do not have free ideal.
LA - eng
KW - real measurable function; lattice-ordered ring; realcompact measurable space; real Riesz map; free ideal
UR - http://eudml.org/doc/299132
ER -