Riesz-Kakutani Representation Theorem

Theorem
Let $X$ be a Hausdorff compact space.

Let $\map \BB X$ be the Borel $\sigma$-algebra on $X$.

Let $\map C {X, \R}$ be the space of real-valued continuous functions.

Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $X$.

Let $\struct {C', \norm {\, \cdot \,}_{C'} }$ be the normed dual space of $\struct {\map C {X, \R}, \norm {\, \cdot \,}_\infty}$.

Then $\ell \in C'$ there is a unique signed measure $\mu$ on $\struct {X, \map \BB X}$ such that:
 * the variation $\size \mu$ of $\mu$ is finite
 * $\ds \forall f \in \map C {X, \R} : \map \ell f = \int_X f \rd \mu$

In addition:
 * $\norm \ell_{C'} = \map {\size \mu} X$