Order-theoretic properties of some sets of quasi-measures

@article{Lipecki2017,
abstract = {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 $\operatorname\{extr\} E(\mu )$.},
author = {Lipecki, Zbigniew},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {linear lattice; ideal; order bounded; ideal dominated; order unit; Banach lattice; $\textit \{AM\}$-space; convex set; extreme point; weakly compact; additive set function; quasi-measure; atomic; extension},
language = {eng},
number = {2},
pages = {197-212},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Order-theoretic properties of some sets of quasi-measures},
url = {http://eudml.org/doc/288211},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Lipecki, Zbigniew
TI - Order-theoretic properties of some sets of quasi-measures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 2
SP - 197
EP - 212
AB - 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 $\operatorname{extr} E(\mu )$.
LA - eng
KW - linear lattice; ideal; order bounded; ideal dominated; order unit; Banach lattice; $\textit {AM}$-space; convex set; extreme point; weakly compact; additive set function; quasi-measure; atomic; extension
UR - http://eudml.org/doc/288211
ER -