measure

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

1. $\mu (A)\ge 0$ for $A\in ℬ(E)$, 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 $A_i\in ℬ(E)$.

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 non-negative extended reals.

The second property above is called countable additivity, or $\sigma$-additivity. A finitely additive measure $\mu$ has the same definition except that $ℬ(E)$ 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 $(E,ℬ(E),\mu )$ 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 ${ℝ}^n$ is one important example of a measure.

Title	measure
Canonical name	Measure
Date of creation	2013-03-22 11:57:33
Last modified on	2013-03-22 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