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
\[\mu(E) = \sup\bracs{\mu(K)| K \subset E, K \text{ compact}}\]
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