PlanetMath: $\sigma$-finite

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$\sigma$ -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) ]

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