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60A10-Measure.tex
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60A10-Measure.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{Measure}
\pmcreated{2013-03-22 11:57:33}
\pmmodified{2013-03-22 11:57:33}
\pmowner{djao}{24}
\pmmodifier{djao}{24}
\pmtitle{measure}
\pmrecord{19}{30756}
\pmprivacy{1}
\pmauthor{djao}{24}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{60A10}
\pmclassification{msc}{28A10}
\pmrelated{LpSpace}
\pmrelated{SigmaFinite}
\pmrelated{Integral2}
\pmrelated{Distribution}
\pmrelated{LebesgueMeasure}
\pmdefines{measure space}
\pmdefines{probability space}
\pmdefines{probability measure}
\pmdefines{countably additive}
\pmdefines{finitely additive}
\pmdefines{$\sigma$-additive}
\pmdefines{positive measure}
\endmetadata
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%%%\usepackage{xypic}
\newcommand{\union}{\cup}
\begin{document}
Let $(E, \mathcal{B}(E))$ be a measurable space. A \emph{measure} on $(E,\mathcal{B}(E))$ is a function $\mu\colon \mathcal{B}(E) \to \mathbb{R} \union \{\infty\}$ with values in the extended real numbers such that:
\begin{enumerate}
\item $\mu(A) \geq 0$ for $A \in \mathcal{B}(E)$, with equality if $A = \emptyset$
\item $\mu(\bigcup_{i=0}^\infty A_i) = \sum_{i=0}^\infty \mu(A_i)$ for any sequence of pairwise disjoint sets $A_i \in \mathcal{B}(E)$.
\end{enumerate}
Occasionally, the term \emph{positive measure} is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.
The second property above is called countable additivity, or $\sigma$-additivity. A \emph{finitely additive measure} $\mu$ has the same definition except that $\mathcal{B}(E)$ is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.
The triple $(E, \mathcal{B}(E), \mu)$ is called a \emph{measure space}. If $\mu(E) = 1$, then it is called a \emph{probability space}, and the measure $\mu$ is called a \emph{probability measure}.
Lebesgue measure on $\mathbb{R}^n$ is one important example of a measure.
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\end{document}