---- > [!definition] Definition. ([[locally finite measure]]) > Let $(X, \tau)$ be a [[Hausdorff space|Hausdorff]] [[topological space]]. Suppose $\Sigma$ is a [[σ-algebra|sigma algebra]] on $X$ satisfying $\Sigma \supset \tau$. A [[measure]] $\mu$ on $\Sigma$ is called **locally finite** if for all $x \in X$, there exists an open [[neighborhood]] $U$ of $x$ for which $\mu(U) < \infty$. ^definition > [!equivalence] > $\mu$ is locally finite if and only if $\mu(K)<\infty$ for every [[compact]] set $K$. ^equivalence ---- #### ---- #### 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 > ```