| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-08-15T13:36:56Z |
| Last updated:
|
2024-02-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151795 |
| . |
| Reference:
|
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