Sigma-finite measure

A measure space $(X,\Sigma,\mu)$ is sigma-finite if there are measurable sets $E_1,E_2,\ldots$ with $X=\bigcup_jE_j$ and $\mu(E_j)<\infty$ for every $j$ .

Examples and use

Lebesgue measure on $\mathbb R^n$ is sigma-finite: use bounded cubes of increasing size. A finite measure is sigma-finite. Counting measure on an uncountable set is not, since a countable union of finite sets is countable. Sigma-finiteness is a hypothesis of the standard product-measure uniqueness and Tonelli statements; it does not mean that the whole space has finite measure.