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# 7: Volume and Measure

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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.

7: Volume and Measure is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.