PlanetMath: $\sigma$-finite

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (16)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

$\sigma$ -finite

(Definition)

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

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

(

is a cube with center at 0 and side length

, and its measure is

), but $\mu(\mathbb{R}^n)=\infty$ .

" $\sigma$ -finite" is owned by Koro. [ full author list (2) | owner history (1) ]

(view preamble)