SignedMeasure 2003-02-10 17:03:17 2005-02-25 17:55:48 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. \usepackage{mathrsfs} % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{signed measure} on a measurable space $(\Omega,\mathscr{S})$ is a function $\mu:\mathscr{S}\rightarrow \mathbb{R}\cup\{+\infty\}$ 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. For some authors, a signed measure can only take finite values (so that $+\infty$ is not allowed either). This is sometimes useful because it turns the space of all signed measures into a normed vector space, with the natural operations, and the norm given by $\|\mu\| = |\mu|(\Omega)$. \item An important example of signed measures arises from the usual measures in the following way: Let $(\Omega,\mathscr{S},\mu)$ 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}