2015
AUTHORSPiermarco Cannarsa , Teresa D’Aprile
ABSTRACTThe 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... »
PAGES3-35
Introduction to Measure Theory and Functional Analysis
ISBN
978-3-319-17018-3
978-3-319-17019-0
http://scigraph.springernature.com/pub.10.1007/978-3-319-17019-0_1
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