$\sigma$-finite

A measure space $(\mathrm{\Omega },ℬ,\mu )$ is a finite measure space if ; 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 $ℱ\subset ℬ$ such that for each $A\in ℱ$, and$\mathrm{\Omega }={\bigcup }_{A\in ℱ}A.$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 ${ℝ}^n$: it is $\sigma$-finite but not finite. In fact

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

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