Math 6210 • Real Analysis

Let \(X\) be a compact metric space, \(\mathcal{M}\) the Borel \(\sigma\)-algebra on \(X\), and \(\mu\) a
finite measure on \(\mathcal{M}\) (meaning that \(\mu(X) < \infty\); we call such a measure a
finite Borel measure on \(X\)). Prove that \(\mu\) is regular, i.e. for every Borel set \(E \subset X\):
\begin{itemize}
\item \(\mu(E) = \sup\{\mu(K) \mid K \subseteq E \text{ is compact}\}\), and
\item \(\mu(E) = \inf\{\mu(U) \mid U \supseteq E \text{ is open}\}\).
\end{itemize}

Hint: Show that the set of \(E\)'s that satisfy both bullets is a \(\sigma\)-algebra
that contains compact sets.