Riesz-Markov-Kakutani Representation Theorem/Construction of mu and M

Construction of $\mu$ and $\MM$
For every $V \in \tau$, define:
 * $\map {\mu_1} V = \sup \set {\Lambda f: f \prec V}$

Note that $\mu_1$ is monotonically increasing.

That is, for all $V, W \in \tau$ such that $V \subset W$, we have:

For every other subset $E \subset X$, define:
 * $\map \mu E = \inf \set {\map {\mu_1} V: V \supset E \land V \in \tau}$

Since $\mu_1$ is monotonically increasing:
 * $\mu_1 = \mu {\restriction_\tau}$

Define:
 * $\MM_F = \set {E \subset X : \map \mu E < \infty \land \map \mu E = \sup \set {\map \mu K: K \subset E \land K \text { compact} } }$

Define:
 * $\MM = \set {E \subset X : \forall K \subset X \text { compact}: E \cap K \in \MM_F}$