# outer regular

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

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

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

 $\mu(B)=\sup\,\{\mu(K)\mid K\subset B,\ K\ \rm{compact}\}.$
Title outer regular OuterRegular 2013-03-22 12:39:57 2013-03-22 12:39:57 djao (24) djao (24) 4 djao (24) Definition msc 28A12 BorelSigmaAlgebra inner regular