18.5 Regular Measures

Definition 18.5.1 (Inner Regular).label 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 18.5.2 (Outer Regular).label 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 18.5.3 (Regular).label 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 18.5.4 ([Theorem 7.8, Fol99]).label Let $X$ be a topological space and $\mu: \cb_{X} \to [0, \infty]$ be a Borel measure. If:

1. (a)

   $X$ is a LCH space.

2. (b)

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

3. (c)

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

then $\mu$ is a regular measure.