measure
Let $(E,\mathcal{B}(E))$ be a measurable space^{}. A measure^{} on $(E,\mathcal{B}(E))$ is a function $\mu :\mathcal{B}(E)\to \mathbb{R}\cup \{\mathrm{\infty}\}$ with values in the extended real numbers such that:

1.
$\mu (A)\ge 0$ for $A\in \mathcal{B}(E)$, with equality if $A=\mathrm{\varnothing}$

2.
$\mu ({\bigcup}_{i=0}^{\mathrm{\infty}}{A}_{i})={\sum}_{i=0}^{\mathrm{\infty}}\mu ({A}_{i})$ for any sequence of pairwise disjoint sets ${A}_{i}\in \mathcal{B}(E)$.
Occasionally, the term positive measure is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the nonnegative extended reals.
The second property above is called countable additivity^{}, or $\sigma $additivity. A 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 measure space. If $\mu (E)=1$, then it is called a probability space, and the measure $\mu $ is called a probability measure.
Lebesgue measure^{} on ${\mathbb{R}}^{n}$ is one important example of a measure.
Title  measure 
Canonical name  Measure 
Date of creation  20130322 11:57:33 
Last modified on  20130322 11:57:33 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  19 
Author  djao (24) 
Entry type  Definition 
Classification  msc 60A10 
Classification  msc 28A10 
Related topic  LpSpace 
Related topic  SigmaFinite 
Related topic  Integral2 
Related topic  Distribution^{} 
Related topic  LebesgueMeasure 
Defines  measure space 
Defines  probability space 
Defines  probability measure 
Defines  countably additive 
Defines  finitely additive 
Defines  $\sigma $additive 
Defines  positive measure 