Riesz-Markov-Kakutani Representation Theorem/Lemma 6

Lemma
For all $E \in \MM_F$ and $\epsilon \in \R_{>0}$, there exist some compact $K \in X$ and some $V \in \tau$ such that:
 * $K \subset E \subset V$

and:
 * $\map \mu {V \setminus K} < \epsilon$

Proof
Let $E \in \MM_F$.

Then, by definition of $\mu$ and $\MM_F$, there exist compact $K \subset E$ and open $V \supset E$ such that:
 * $\map \mu V - \dfrac \epsilon 2 < \map \mu E < \map \mu K + \dfrac \epsilon 2$

By Compact Subspace of Hausdorff Space is Closed:
 * $V \setminus K \in \tau$

So, by Lemma 5:
 * $V \setminus K \in \MM_F$

By Lemma 4:
 * $\map \mu {V \setminus - K} = \map \mu V - \map \mu K < \epsilon$