Riesz-Markov-Kakutani Representation Theorem/Lemma 5

Lemma
$\set {V \in \tau: \map \mu V < \infty} \subset \MM_F$

Proof
Since $\mu$ and $\mu_1$ coincide on $\tau$, by definition of $\mu_1$, for all $\alpha < \map \mu V$, there exists some $f \in \map {C_c} X: f \prec V$ such that $\Lambda f > \alpha$.

Let $K = \supp f$.

For all $W \in \tau$ such that $K \subset W$:
 * $f \prec W$

So, by definition of $\mu_1$:
 * $\Lambda f \le \map \mu W$

By definition of $\mu$:
 * $\Lambda f \le \map \mu K$

So:
 * $\alpha < \map \mu K$

That is, for all $\alpha < \map \mu V$, there exists some compact $K \subset V$ such that:
 * $\alpha < \map \mu K \le \map \mu V$

Since $V$ was arbitrary:
 * $\set {V \in \tau: \map \mu V < \infty} \subset \MM_F$