| Title:
|
Order-theoretic properties of some sets of quasi-measures (English) |
| Author:
|
Lipecki, Zbigniew |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
197-212 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathfrak M$ and $\mathfrak R$ be algebras of subsets of a set $\Omega $ with $\mathfrak M\subset \mathfrak R$, and denote by $E(\mu )$ the set of all quasi-measure extensions of a given quasi-measure $\mu $ on $\mathfrak M$ to $\mathfrak R$. We show that $E(\mu )$ is order bounded if and only if it is contained in a principal ideal in $ba(\mathfrak R)$ if and only if it is weakly compact and $\operatorname{extr} E(\mu )$ is contained in a principal ideal in $ba(\mathfrak R)$. We also establish some criteria for the coincidence of the ideals, in $ba(\mathfrak R)$, generated by $E(\mu )$ and $\operatorname{extr} E(\mu )$. (English) |
| Keyword:
|
linear lattice |
| Keyword:
|
ideal |
| Keyword:
|
order bounded |
| Keyword:
|
ideal dominated |
| Keyword:
|
order unit |
| Keyword:
|
Banach lattice |
| Keyword:
|
$\textit{AM}$-space |
| Keyword:
|
convex set |
| Keyword:
|
extreme point |
| Keyword:
|
weakly compact |
| Keyword:
|
additive set function |
| Keyword:
|
quasi-measure |
| Keyword:
|
atomic |
| Keyword:
|
extension |
| MSC:
|
06F20 |
| MSC:
|
28A12 |
| MSC:
|
28A33 |
| MSC:
|
46A55 |
| MSC:
|
46B42 |
| idZBL:
|
Zbl 06773714 |
| idMR:
|
MR3666941 |
| DOI:
|
10.14712/1213-7243.2015.208 |
| . |
| Date available:
|
2017-06-13T13:23:17Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146788 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
