Measure Spaces View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2015

AUTHORS

Piermarco Cannarsa , Teresa D’Aprile

ABSTRACT

The concept of measure of a set originates from the classical notion of volume of an interval in \(\mathbb R^N\). Starting from such an intuitive idea, by a covering process one can assign to any set a nonnegative number which “quantifies its extent”. Such an association leads to the introduction of a set function called exterior measure, which is defined for all subsets of \(\mathbb R^N\). The exterior measure is monotone but fails to be additive. Following Carathéodory’s construction, it is possible to select a family of sets for which the exterior measure enjoys further properties such as countable additivity. By restricting the exterior measure to such a family one obtains a complete measure. This is the procedure that allows to define the Lebesgue measure in \(\mathbb R^N\). The family of all Lebesgue measurable sets is very large: sets that fail to be measurable can only be constructed by using the Axiom of Choice. More... »

PAGES

3-35

Book

TITLE

Introduction to Measure Theory and Functional Analysis

ISBN

978-3-319-17018-3
978-3-319-17019-0

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-17019-0_1

DOI

http://dx.doi.org/10.1007/978-3-319-17019-0_1

DIMENSIONS

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