Riesz-Markov-Kakutani Representation Theorem/Lemma 4

Lemma
$\mu$ is countably additive over pairwise disjoint collections of subsets of $\MM_F$.

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

Let $\map \mu E = \infty$.

Then, by countable subadditivity:


 * $\ds \infty = \map \mu E \le \sum_{i \mathop = 1}^\infty \map \mu {E_i}$

So:
 * $\ds \map \mu E = \sum_{i \mathop = 1}^\infty \map \mu E$

Suppose $\map \mu E < \infty$.

By definition of $\MM_F$, for all $\epsilon \in \R_{>0}$, for each $i$, there exists a compact $H_i \subset E_i$ such that:
 * $\map \mu {H_i} > \map \mu {E_i} - 2^{-i} \epsilon$

So:

This holds for all $n \in \N$.

So $\mu$ is countably superadditive over pairwise disjoint collections of subsets of $\MM_F$.

Therefore, by Lemma 1, $\mu$ is countably additive over pairwise disjoint collections of subsets of $\MM_F$.