In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.

Let ${\displaystyle (X,T)}$ be a Hausdorff topological space and let ${\displaystyle \Sigma }$ be a ${\displaystyle \sigma }$-algebra on ${\displaystyle X}$ that contains the topology ${\displaystyle T}$ (so that every open set is a measurable set, and ${\displaystyle \Sigma }$ is at least as fine as the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle X}$). A measure/signed measure/complex measure ${\displaystyle \mu }$ defined on ${\displaystyle \Sigma }$ is called locally finite if, for every point ${\displaystyle p}$ of the space ${\displaystyle X,}$ there is an open neighbourhood ${\displaystyle N_{p}}$ of ${\displaystyle p}$ such that the ${\displaystyle \mu }$-measure of ${\displaystyle N_{p}}$ is finite.

In more condensed notation, ${\displaystyle \mu }$ is locally finite if and only if $${\displaystyle {\text{for all }}p\in X,{\text{ there exists }}N_{p}\in T{\mbox{ such that }}p\in N_{p}{\mbox{ and }}\left|\mu \left(N_{p}\right)\right|<+\infty .}$$