Locally finite measure

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

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

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

Examples

 * 1) Any probability measure on $$X$$ is locally finite, since it assigns unit measure to the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite.
 * 2) Lebesgue measure on Euclidean space is locally finite.
 * 3) By definition, any Radon measure is locally finite.
 * 4) The counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line with its usual Borel topology is not.