---- > [!definition] Definition. ([[finite measure]]) > Let $\mu:\Sigma \to [0, \infty]$ be a [[measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. > We say $\mu$ is **finite** if $\mu(X)<\infty$. > More weakly, we say $\mu$ is **$\sigma$-finite** if $X$ can be written as a [[countably infinite|countable union]] of sets with finite measure. In other words, $\mu$ is $\sigma$-finite if there exists a [[sequence]] $X_{1},X_{2},\dots$ of sets in $\Sigma$ such that $X=\bigcup_{k=1}^{\infty} X_{k} \text{ and } \mu(X_{k})<\infty.$ ^definition - [ ] relation to [[locally finite measure]] > [!basicexample] > The [[Lebesgue measure]] on $[0,1]$ is a finite measure. The [[Lebesgue measure]] on $\mathbb{R}$ is not finite, but it *is* $\sigma$-finite. The [[measure|counting measure]] on $\mathbb{R}$ is not $\sigma$-finite. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```