Riesz-Markov-Kakutani Representation Theorem/Lemma 3

Lemma
$\mu$ is countably additive over pairwise disjoint collections of compact sets.

Proof
Let $K_1$ and $K_2$ be disjoint compact subsets of $X$.

By Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods, there exists an open $V\supset K_1$ such that $V$ and $K_2$ are disjoint.

By Urysohn's Lemma, there exists a mapping $f \in \map{C_c} X$ such that:
 * $\map f {K_1} = \set 1$

and:
 * $\map f {K_2} = \set 0$

By Lemma 2 and union of compact sets is compact:
 * $\forall \epsilon \in \R_{>0}: \exists g \in \map {C_c} X: K_1 \cup K_2 \prec g \text{ and } \Lambda g < \map \mu {K_1 \cup K_2} + \epsilon$

Now:
 * $K_1 \prec f g$

and:
 * $K_2 = \prec \paren {1 - f} g$

By linearity of $\Lambda$ and Lemma 2:

Thus, by Lemma 1, $\mu$ is additive for disjoint compact sets.

Applying the Principle of Mathematical Induction yields countable additivity.