Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Lipecki, Zbigniew. "Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure." Commentationes Mathematicae Universitatis Carolinae 56.3 (2015): 331-345. <http://eudml.org/doc/271563>.

@article{Lipecki2015,
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 give some criteria for order boundedness of $E(\mu )$ in $ba(\mathfrak \{R\})$, in the general case as well as for atomic $\mu $. Order boundedness implies weak compactness of $E (\mu )$. We show that the converse implication holds under some assumptions on $\mathfrak \{M\}$, $\mathfrak \{R\}$ and $\mu $ or $\mu $ alone, but not in general.},
author = {Lipecki, Zbigniew},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {linear lattice; order bounded; additive set function; quasi-measure; atomic; extension; convex set; extreme point; weakly compact},
language = {eng},
number = {3},
pages = {331-345},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure},
url = {http://eudml.org/doc/271563},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Lipecki, Zbigniew
TI - Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 3
SP - 331
EP - 345
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 give some criteria for order boundedness of $E(\mu )$ in $ba(\mathfrak {R})$, in the general case as well as for atomic $\mu $. Order boundedness implies weak compactness of $E (\mu )$. We show that the converse implication holds under some assumptions on $\mathfrak {M}$, $\mathfrak {R}$ and $\mu $ or $\mu $ alone, but not in general.
LA - eng
KW - linear lattice; order bounded; additive set function; quasi-measure; atomic; extension; convex set; extreme point; weakly compact
UR - http://eudml.org/doc/271563
ER -