Riesz-Markov-Kakutani Representation Theorem

Theorem
Let $\struct{X, \tau}$ be a Hausdorff space.

Let $\map {C_c} {X}$ be the space of continuous complex functions with compact support on $X$.

Let $\Lambda$ be a positive linear functional on $\map {C_c} {X}$.

There exists a $\sigma$-algebra $\MM$ over $X$ which contains the Borel $\sigma$-algebra of $\struct{X, \tau}$.

There exists a unique complete Radon measure $\mu$ on $\MM$ such that
 * $\forall f\in \map{C_c}{X}, \Lambda f = \int_X f \rd \mu$.

Notation
For an open set $V\in\tau$ and a function $f\in\map{C_c}{X}$,


 * $f \prec V \iff \supp f \subset V$.

For a compact set $K\subset X$ and a function $f\in\map{C_c}{X}$,


 * $K \prec f \iff \forall x\in K, f(x)=1$.

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

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

By monotonicity of $\mu_1$, $\mu_1 = \mu {\restriction_\tau}$.

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