Wydawnictwa / Czasopisma IMPAN / Dissertationes Mathematicae / Wszystkie zeszyty

Streszczenie

The memoir is based on a series of six papers by the author published over the years 1995–2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let $\mathfrak M$ and $\mathfrak R$ be algebras of subsets of a set $\Omega$ with $\mathfrak M\subset\mathfrak R$. Given a quasi-measure $\mu$ on ${\mathfrak M}$, i.e., $\mu\in ba_+(\mathfrak M)$, we denote by $E(\mu)$ the convex set of all quasi-measure extensions of $\mu$ to $\mathfrak R$. Moreover, we denote by $s$, $w$ and $w^*$ the strong, weak and weak$^*$ topologies of the dual Banach lattice $ba(\mathfrak R)$, respectively. Our starting point are the following two properties of $E(\mu)$ and $\mathop{\rm extr}E(\mu)$, which are easy consequences of known results:

(a) $(E(\mu ),w^*)$ is compact;

(b) $\mathop{\rm extr}E(\mu)$ is closed in $(ba(\mathfrak R),s)$.

We study the following conditions related to (a) and (b):

(i) $(E(\mu ),s)$ is compact;

(ii) $(E(\mu ),w)$ is compact;

(iii) $s$ and $w$ coincide on $E(\mu )$;

(iv) $s$ and $w$ coincide on $\mathop{\rm extr}E(\mu)$;

(v) $s$ and $w^*$ coincide on $\mathop{\rm extr}E(\mu)$;

(vi) $w$ and $w^*$ coincide on $\mathop{\rm extr}E(\mu)$;

(vii) $\mathop{\rm extr}E(\mu)$ is closed in $(ba(\mathfrak R),w)$;

(viii) $\mathop{\rm extr}E(\mu)$ is closed in $(ba(\mathfrak R),w^*)$;

(ix) $(\mathop{\rm extr}E(\mu),s)$ is compact;

(x) $(\mathop{\rm extr}E(\mu),w)$ is compact;

(xi) $(\mathop{\rm extr}E(\mu),w^*)$ is compact;

(xii) $(\mathop{\rm extr}E(\mu),s)$ is discrete;

(xiii) $(\mathop{\rm extr}E(\mu),w)$ is discrete;

(xiv) $(\mathop{\rm extr}E(\mu),w^*)$ is discrete;

(xv) $\mathop{\rm extr}E(\mu)$ is dense in $(E(\mu ),w)$;

(xvi) $\mathop{\rm extr}E(\mu)$ is dense in $(E(\mu ),w^*)$.

In most cases, we find various equivalent conditions expressed in topological, affine-topological and measure-theoretic terms. To this end, we use, in particular, the antimonogenic component $\mu^{\rm a} $ of $\mu$. (This is the minimal $\nu\in ba_+\mathfrak M$ such that $\nu\leq\mu$ and $E(\mu-\nu)$ is a singleton.) Here are some sample results: (viii) holds if and only if $\mu^{\rm a} $ is atomic; both (xiii) and (xiv) are equivalent to the condition that $\mu^{\rm a}$ have finite range; (xvi) holds if and only if $\mu^{\rm a} $ is nonatomic. One of our main tools is an affine-topological representation of $E(\mu )$ for atomic $\mu$ as the countable Cartesian product of simplex like sets. We also study some other topological properties of $\mathop{\rm extr}E(\mu)$, such as zero-dimensionality and various kinds of connectedness. Some of our results involve the cardinality $\mathfrak m$ of $\mathop{\rm extr}E(\mu)$. In general, there are no restrictions on $\mathfrak m$ except for $\mathfrak m\neq0$. However, if $\mu$ is nonatomic, then $\mathfrak m^{\aleph_0}=\mathfrak m$. The case where $\mathfrak m\leq\aleph_0$ is also thoroughly investigated.