Riesz-Markov-Kakutani Representation Theorem/Lemma 1

Lemma
$\mu$ is countably subadditive.

Proof
Let $V_1$ and $V_2$ be open subsets of $X$.

Let $g \in \map {C_c} X: g \prec V_1 \cup V_2$.

Let $\set {h_1, h_2}$ be a partition of unity subordinate to $\set {V_1, V_2}$.

By linearity of $\Lambda$ and the definition of $\mu$:

Since $g$ was arbitrary, by the definition of $\mu$, $\mu$ is subadditive over $\tau$.

Applying Principle of Mathematical Induction yields the countable subadditivity of $\mu$ over $\tau$.

Let $\sequence{E_i}\in X^\N$.

By definition of $\mu$, for all $E_i$, for all $\epsilon \in \R_{>0}$, there exists an open $V_i \supset E_i$ such that:
 * $\map \mu {V_i} \le \map \mu {E_i} + 2^{-i} \epsilon$

Therefore, by $\ds \bigcup_{i \mathop = 1}^\infty V_i \supset \bigcup_{i \mathop = 1}^\infty E_i$, monotonicity of $\mu$, and countable subadditivity of $\mu$ over $\tau$:
 * $\ds \map \mu {\bigcup_{i \mathop = 1}^\infty E_i} \le \map \mu {\bigcup_{i \mathop = 1}^\infty V_i} \le \epsilon + \sum_{i \mathop = 1}^\infty \map \mu {E_i}$

Since $\epsilon$ was arbitrary, $\mu$ is countably subadditive over $X$.