Regular measure on Borel sets I am trying to do the following problem:
Let $\mu$ be a measure defined on the Borel sets of $\mathbb R^n$ such that $\mu$ takes finite values on the compact sets. Let $\mathcal H$ be the class of Borel sets such $E$  with $$(i) \space \mu(E)=\inf\{\mu(G), E \subset G, G \space \text{open}\},$$$$(ii) \space \mu(E)=\sup\{\mu(K), K \subset E, K \space \text{compact}\}$$
Prove that the open and compact sets are in $\mathcal H$. If $\mu$ is finite, then $\mathcal H$ is a $\sigma-$algebra.
It is clear that if $O$ is open then $O$ satisfies $(i)$, analogously, if $K$ is compact then $K$ satisfies (ii). Now, I have no idea how to show that $O$ and $K$ satisfy (ii) and (i) respectively.
As for the second part, I am pretty lost as well. I am trying to prove that if $A \in \mathcal H$, then $A^c \in \mathcal H$, and that if $(A_i)_{i \in \mathbb N} \subset \mathcal H$, then so is its union.
I can write $\mu(A^c)=\mu(X \setminus A)=\mu(X)-\mu(A)$ (here I use $\mu(X)<\infty$), but I don't know how to use the fact that $X,A \in \mathcal H$.
Any suggestions to do this exercise would be greatly appreciated.
 A: Suggestions:


*

*To show that compact sets satisfy (i), use the metric on $\mathbb R^n$ to construct a decreasing sequence $\{G_n\}$ of open sets whose intersection is the compact set. What can you say about the measures of the $G_n$ and their relation to that of the compact set?

*To show that open sets satisfy (ii), build your approximations using dyadic $n$-cubes $\prod_{i=1}^n \left[\frac{m_i}{2^k}, \frac{m_i + 1}{2^k} \right]$ where $m_i \in \mathbb Z$ and $k \in \mathbb Z_{>0}$.

*When $\mu$ is finite, the following equivalent definition of $\mathcal H$ may be more useful:

$E \in \mathcal H$ iff for every $\epsilon > 0$ there exists an open $G$ and a compact $K$ such that $K \subset E \subset G$ and $\mu(G \setminus K) < \epsilon$.

Closure of $\mathcal H$ under complementation should be essentially straightforward now.
For closure under countable unions, choose for each $E_n$ sufficiently tight $G_n$ and $K_n$, and use them to construct $G$ and $K$ that are of the required tightness around $\bigcup E_n$.
Be careful that in several places, the most "natural" approximations from the inside might not necessarily be closed and/or bounded, hence not compact. A major part of this exercise is finding workarounds for this issue.
