$\sigma$-finiteSigmaFinite2002-02-27 00:29:132006-09-13 22:16:45Definition$\sigma$-infinitesigma-infinitefinite measure space\usepackage{amssymb}
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A measure space $(\Omega, \mathcal{B}, \mu)$ is a \textbf{finite measure space} if $\mu(\Omega)<\infty$; it is $\sigma$-\textbf{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 $\sigma$-\textbf{infinite}.
\textbf{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 $\mu(\mathbb{R}^n)=\infty$.