Signed Measures View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2015

AUTHORS

Piermarco Cannarsa , Teresa D’Aprile

ABSTRACT

Given a measure space \((X,\fancyscript{E}, \mu )\) and a function \(\rho \in L^1(X,\mu )\), the so-called Lebesgue indefinite integral \(\nu (E)=\int _E \rho \,d\mu (E\in \fancyscript{E})\) defines a \(\sigma \)-additive set function, that is, if \(E=\bigcup _n E_n\), with \(E_n\in \fancyscript{E}\) a family of disjoint sets, then \(\nu (E)=\sum _n \nu (E_n).\) Therefore, when \(\rho \ge 0\), \(\nu \) is a finite measure on \(\fancyscript{E}\) satisfying \( E\in \fancyscript{E}\;\; \& \;\; \mu (E)=0\;\;\Rightarrow \;\;\nu (E)=0.\) More... »

PAGES

253-270

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_8

DOI

http://dx.doi.org/10.1007/978-3-319-17019-0_8

DIMENSIONS

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