7: Volume and Measure
( \newcommand{\kernel}{\mathrm{null}\,}\)
I. Our theory of set families leads quite naturally to a generalization of metric spaces. As we know, in any such space (S,ρ), there is a family G of open sets, and a family F of all closed sets. In Chapter 3, §12, we derived the following two properties.
(i) G is closed under any (even uncountable) unions and under finite intersections (Chapter 3, §12, Theorem 2). Moreover,
∅∈G and S∈G.
(ii) F has these properties, with "unions" and "intersections" interchanged (Chapter 3, §12, Theorem 3). Moreover, by definition,
A∈F iff −A∈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, G and F, with properties (i) and (ii), in a suitable manner. For examples, see Problems 1 to 4 below. Once G and 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.