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'signed measure'
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| Title of object: |
signed measure |
| Canonical Name: |
SignedMeasure |
| Type: |
Definition |
| Created on: |
2003-02-10 17:03:17.25733-05 |
| Modified on: |
2003-02-10 18:10:16.562148-05 |
| Classification: |
msc:28A12 |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
A \emph{signed measure} on a measurable space $(\Omega,\mathcal{A})$ is a function $\mu:\mathcal{A}\rightarrow \mathbb{R}\cup\{+\infty\}$ which is \PMlinkname{$\sigma$-additive}{Additive} and such that $\mu(\emptyset)=0$.
\textbf{Remarks.}
\begin{enumerate}
\item The usual (positive) measure is a particular case of signed measure, in which $|\mu| = \mu$ (see Jordan decomposition.)
\item Notice that the value $-\infty$ is not allowed.
\item An important example of signed measures arises from the usual measures in the following way: Let $(\Omega,\mathcal{A},\mu)$ be a measure space, and let $f$ be a (real valued) measurable function such that
\[\int_{\{x\in \Omega:f(x)<0\}} |f| d\mu <\infty.\]
Then a signed measure is defined by
\[A\mapsto \int_A fd\mu.\]
\end{enumerate} |
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