Riesz-Markov-Kakutani Representation Theorem/Lemma 8

Lemma
$\MM_F$ is closed under set difference, union and intersection.

Proof
Let $\tuple {A, B} \in \paren {\MM_F}^2$.

By Lemma 6, there exist compact sets $K_1, K_2$ and open sets $V_1, V_2$ such that:
 * $K_1 \subset A \subset V_1$
 * $K_2 \subset B \subset V_2$

and:
 * $\forall i \in \set {1, 2}: \map \mu {V_i \setminus K_i} < \dfrac \epsilon 2$

Now:

So, by Lemma 1:
 * $\map \mu {A_B} \le \map \mu {K_1 \setminus V_2} + \epsilon$

By Closed Subspace of Compact Space is Compact:
 * $K_1 \setminus V_2$ is compact.

Since $K_1 \setminus V_2 \subset A \setminus B$, there exist compact subsets of $A \setminus B$ arbitrarily close in measure to $A \setminus B$.

So:
 * $A \setminus B \in \MM_F$.

Now, by Lemma 7:
 * $A \cup B = \paren {A \setminus B} \cup B \in \MM_F$

and:
 * $A \cap B = A \setminus \paren {A \setminus B} \in \MM_F$.