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Mathematics LibreTexts

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,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 SG.

(ii) F has these properties, with "unions" and "intersections" interchanged (Chapter 3, §12, Theorem 3). Moreover, by definition,

AF iff AG.

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.


This page titled 7: Volume and Measure is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform.

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