Riesz Theorem on C[K], K compact I'm studying Riez Theorem on Kreyszig's book: "Introductory functional analysis" , it states that
"Let $l$ a bounded and linear functional on $C[a,b]$ (continuous functions on [a,b]) , then $l$ can be represented as a Rieman-Stieltjes integral
$$l(f)=\int_a^bf(x)d\mu(x)$$
where $\mu$ is of bounded variation on $[a,b]$"
The prove is very clear there, my question is, can I extend this prove to $C(K)$ where K is any compact in $\mathbb R$?. 
I know the general Riez-Markov theorem for any Compact Hausdorff Space, but the proof of this is very complicate and this result is bigger than I need. Thank you, any help would be appreciate.
 A: Let $[a,b]$ be the smallest containing interval for the compact set $K$. Then $K\setminus[a,b]$ is an open subet of $\mathbb{R}$ and, as such, consists of an empty, finite, or countably infinite set of disjoint open intervals $\{ I_j \}$. So every function $f\in C(K)$ can be extended linearly on each $I_j$ in order to obtain a unique $\tilde{f} \in C[a,b]$ such that $\|\tilde{f}\|_{C[a,b]}=\|f\|_{C(K)}$. It seems to me this extension operator is linear. That is, if $E(f)=\tilde{f}$, then $E : C(K)\mapsto C[a,b]$ is linear. And $E$ is isometric because the sup norm is never increased by such an extension, and it's certainly not smaller than the sup norm for the original $f$.
Every continuous linear functional $\Phi$ on $C(K)$ is a continuous linear functional on $E(C(K))\subset C[a,b]$ and extends continuously to its closure $\overline{E(C(X))}$ in $C[a,b]$. Then, by the Hahn-Banach Theorem, $\Phi$ extends to a continuous linear functional $\tilde{\Phi}$ on $C[a,b]$. Finally,
$$
           \Phi(f)= \tilde{\Phi}(Ef) = \int_{a}^{b}\tilde{f}(t)d\rho(t).
$$
I'm stuck at this point because I cannot see how to argue that $\rho$ is constant on the open intervals $\{I_j\}$, even though I suspect there must be some way to arrange for this to hold by requiring certain properties for the extension, if it isn't generally true.
