Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

We show that the cardinality ${\mathfrak n}$ of a compact convex set $W$ in a topological linear space $X$ satisfies the condition that ${\mathfrak n}^{\aleph_0} = {\mathfrak n}$. We also establish some relations between the cardinality of $W$ and that of $\mathop{\rm extr}\nolimits{W}$ provided $X$ is locally convex. Moreover, we deal with the cardinality of the convex set $E(\mu)$ of all quasi-measure extensions of a quasi-measure $\mu$, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of $\mathop{\rm extr}\nolimits E(\mu)$.