14.5 Regular Measures

Definition 14.5.1 (Inner Regular). Let $X$ be a topological space, $\mu: \cb_{X} \to [0, \infty]$ be a Borel measure, and $E \in \cb_{X}$, then $\mu$ is inner regular on $E$ if

Definition 14.5.2 (Outer Regular). Let $X$ be a topological space, $\mu: \cb_{X} \to [0, \infty]$ be a Borel measure, and $E \in \cb_{X}$, then $\mu$ is outer regular on $E$ if

Definition 14.5.3 (Regular). Let $X$ be a topological space and $\mu: \cb_{X} \to [0, \infty]$ be a measure, then $\mu$ is regular if it is inner regular and outer regular on all Borel sets.

Theorem 14.5.4. Let $X$ be a topological space and $\mu: \cb_{X} \to [0, \infty]$ be a Borel measure. If:

1. $X$ is a LCH space.

2. Every open set of $X$ is $\sigma$-compact.

3. For any $K \subset X$ compact, $\mu(K) < \infty$.

then $\mu$ is a regular measure.