$\sigma$ -finite

A measure space $(\Omega,\mathcal{B},\mu)$ is a finite measure space if $\mu(\Omega)<\infty$ ; it is $\sigma$ -finite if the total space is the union of a finite or countable family of sets of finite measure, i.e. if there exists a countable set $\mathcal{F}\subset\mathcal{B}$ such that $\mu(A)<\infty$ for each $A\in\mathcal{F}$ , and $\Omega=\bigcup_{A\in\mathcal{F}}A.$ In this case we also say that $\mu$ is a $\sigma$ -finite measure. If $\mu$ is not $\sigma$ -finite, we say that it is $\sigma$ -infinite.

Examples. Any finite measure space is $\sigma$ -finite. A more interesting example is the Lebesgue measure $\mu$ in $\mathbb{R}^{n}$ : it is $\sigma$ -finite but not finite. In fact

\mathbb{R}^{n}=\bigcup_{k\in\mathbb{N}}[-k,k]^{n}

( $[-k,k]^{n}$ is a cube with center at $0$ and side length $2k$ , and its measure is $(2k)^{n}$ ), but $\mu(\mathbb{R}^{n})=\infty$ .

Title	$\sigma$ -finite
Canonical name	sigmafinite
Date of creation	2013-03-22 12:29:48
Last modified on	2013-03-22 12:29:48
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	13
Author	Koro (127)
Entry type	Definition
Classification	msc 28A10
Synonym	$\sigma$ finite
Synonym	sigma-finite
Synonym	sigma finite
Related topic	Measure
Related topic	MeasureSpace
Related topic	AlternativeDefinitionOfSigmaFiniteMeasure
Related topic	AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure
Defines	$\sigma$ -infinite
Defines	sigma-infinite
Defines	finite measure space

σ-finite

$\sigma$ -finite