outer regular

Let $X$ be a locally compact Hausdorff topological space with Borel $\sigma$–algebra $ℬ$, and suppose $\mu$ is a measure on $(X,ℬ)$. For any Borel set $B\in ℬ$, the measure $\mu$ is said to be outer regular on $B$ if

$$\mu (B)=inf\{\mu (U)\mid U\supset B,U\mathrm{open}\}.$$

We say $\mu$ is inner regular on $B$ if

$$\mu (B)=sup\{\mu (K)\mid K\subset B,K\mathrm{compact}\}.$$