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