Riesz-Markov-Kakutani representation theorem

Every positive linear functional $\phi$ on $C(K)$ for a compact Hausdorff space $K$ has a unique representationby a finite regular positive Borel measure $\mu$. More generally, $C(K{)}^{\ast }$ is isometrically the space of finite regular complex Borel measures with the total-variation norm.

A positive linear functional on the space of continuous compactly supported functions of a locally compact Hausdorff space has a unique representing positive Radon measure. Positivity gives boundedness on fixed compact supports: choose a compactly supported continuous cutoff $p\ge 1$ on the support $K$, then $|I(f)|\le \Vert f{\Vert }_{\infty }I(p)$ for real $f$ supported in $K$. The resulting measure is finite on compact sets but need not have finite total mass.

Extend the cylinder premeasure from a positive functional to a Borel probability measure and use cylinder-function density in Cantor space. The integral and functional agree on cylinder simple functions, and both are continuous in the supremum norm, so they agree on all of $C(\Omega )$. Uniqueness on the generating cylinder algebra gives uniqueness of the Borel measure.

Given a continuous surjection $h:\Omega \to K$ from Cantor space to a nonempty compact metric space, the pullback $f\mapsto f\circ h$ is a unital isometric embedding of $C(K)$ into $C(\Omega )$. Transport a normalized positive linear functional to its image, take a positive extension from a unital subspace of C(K), and represent that extension on Cantor space. The pushforward measure $h_{\ast }P$ then represents the original functional.

A finite regular complex measure of total variation one can be approximated on a finite-dimensional vector subspace of continuous functions by a finite sum of phased point evaluations with ${\sum }_i|t_i|=1$. The Krein-Milman theorem and the extreme points of the dual unit ball of C(K) give weak-star approximation by convex combinations of phased point masses. A finite norm net of the vector subspace unit ball turns finitely many scalar approximations into one uniform estimate. Repeated nodes may be kept separate to retain the exact coefficient-magnitude sum.