Riesz-Markov-Kakutani Representation Theorem/Lemma 7

Lemma
The union, if of finite measure, of countable pairwise disjoint subsets of $\MM_F$ is in $\MM_F$.

Proof
Let $\sequence {E_i} \in \paren {\MM_F}^\N$ be pairwise disjoint.

Suppose $\ds E = \bigcup_{i \mathop = 1}^\infty E_i$ has finite measure.

By Lemma 4, for all $\epsilon > 0$, there exists some $N \in \N$ such that:
 * $\ds \map \mu E \le \epsilon \sum_{i \mathop = 1}^\infty \map \mu {E_i}$

By definition of $\MM_F$, for all $i$, there exists some compact $H_i \subset E_i$ such that:
 * $\map \mu {H_i} > \map \mu {E_i} - 2^{-i} \epsilon$

Then:
 * $\ds \map \mu E \le \map \mu {\bigcup_{i \mathop = 1}^N H_i} + 2 \epsilon$

By Finite Union of Compact Sets is Compact:
 * $\ds \bigcup_{i \mathop = 1}^N \subset E$

Therefore:
 * $E \in \MM_F$