7: Volume and Measure
I. Our theory of set families leads quite naturally to a generalization of metric spaces. As we know, in any such space \((S, \rho),\) there is a family \(\mathcal{G}\) of open sets, and a family \(\mathcal{F}\) of all closed sets. In Chapter 3, §12, we derived the following two properties.
(i) \(\mathcal{G}\) is closed under any (even uncountable) unions and under finite intersections (Chapter 3, §12, Theorem 2). Moreover,
\[\emptyset \in \mathcal{G} \text { and } S \in \mathcal{G}.\]
(ii) \(\mathcal{F}\) has these properties, with "unions" and "intersections" interchanged (Chapter 3, §12, Theorem 3). Moreover, by definition,
\[A \in \mathcal{F} \text { iff }-A \in \mathcal{G}.\]
Now, quite often, it is not so important to have distances (i.e., a metric) defined in \(S,\) but rather to single out two set families, \(\mathcal{G}\) and \(\mathcal{F},\) with properties (i) and (ii), in a suitable manner. For examples, see Problems 1 to 4 below. Once \(\mathcal{G}\) and \(\mathcal{F}\) are given, one does not need a metric to define such notions as continuity, limits, etc. (See Problems 2 and 3.) This leads us to the following definition.