5.4 Regular Measures

Definition 5.4.1 (Inner Regular).label Let $X$ be a topological space, $\mu:\cali{B}_{X}\to [0,\infty]$ be a Borel measure, and $E\in\cali{B}_{X}$ then $\mu$ is inner regular on $E$ if

Definition 5.4.2 (Outer Regular).label Let $X$ be a topological space, $\mu:\cali{B}_{X}\to [0,\infty]$ be a Borel measure, and $E\in\cali{B}_{X}$ then $\mu$ is outer regular on $E$ if

Definition 5.4.3 (Regular).label Let $X$ be a topological space and $\mu:\cali{B}_{X}\to[0,\infty]$ be a Borel measure, then $\mu$ is regular if it is inner regular and outer regular on all Borel sets.

Theorem 5.4.4 ([Theorem 7.8, Fol99]).label Let $X$ be a topological space and $\mu:\cali{B}_{X}\to[0,\infty]$ be a Borel measure. If:

1. (1)

   $X$ is an LCH space.

2. (2)

   Every open set of $X$ is $\s$-compact

3. (3)

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

then $\mu$ is a regular measure.