# $\sigma$-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 $\sigma$-.

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 sigmafinite 2013-03-22 12:29:48 2013-03-22 12:29:48 Koro (127) Koro (127) 13 Koro (127) Definition msc 28A10 $\sigma$ finite sigma-finite sigma finite Measure MeasureSpace AlternativeDefinitionOfSigmaFiniteMeasure AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure $\sigma$-infinite sigma-infinite finite measure space