I had the following idea of how to give a simple proof of the Riesz-Markov-Kakutani representation theorem for subsets of the complex numbers and I wanted to know whether someone sees some principle problem. Note that I am a Physicist, thus my knowledge of measure theory is very bad.
If I have given a compact subset $S$ of the complex numbers and a continuous positive linear functional $\phi: C(S) \rightarrow \mathbb{R}$, where $C(S)$ are the continuous function on S. I want to prove that there exist a measure $\nu$, such that $\phi(f) = \int_S f d\nu$. To start with, I want to make S to a measurable set. The measure induced by the lebesgue measure should give me a measure $\mu$ on S. Now continuous functions on that set are dense in $L^2(S,\mu)$. Thus I can extend $\phi$ to a continous positive linear functional on $L^2(S,\mu)$ by the B.L.T. theorem. By the Riesz-representation theorem in Hilbert spaces and $L^2(S,\mu)$ being a Hilbert space, there exist an $g \in L^2(S,\mu)$ such that $\phi(f) = \langle g,f \rangle = \int_S \bar g f d\mu$. Since $\phi$ is positive and continuous, $g$ has to be real and essentially bounded. Then I define $d\nu = g d\mu$ and think of having found my measure.
My question is, does the above logic work and is $d\nu = g d\mu$ really a measure?
Thanks!
It is not true that every continuous linear functional on $C(S)$ can be extended to a continuous linear functional on $L^2(S,\mu)$. The problem is that the notions of continuity on the two spaces are different, since $C(S)$ and $L^2(S,\mu)$ have different topologies.
To be more precise, let $\|\cdot\|_0$ be the supremum norm on $C(S)$ and $\|\cdot\|_2$ be the $L^2$-norm on $L^2(S,\mu)$. Now, let $\phi$ be a $\|\cdot\|_0$-continuous linear functional on $C(S)$. Even if you regard $C(S)$ a subset of $L^2(S,\mu)$, there is no guarantee that $\phi$ is $\|\cdot\|_2$-continuous on $C(S)$.
A simple counterexample is as follows. Assume (with no loss of generality) that $0\in S$ is an interior point. The evaluation at $0$ is a $\|\cdot\|_0$-continuous linear functional (represented by the Dirac mass $\delta_0$). This functional is however not $\|\cdot\|_2$-continuous. Indeed let $(f_n)_n$ be a sequence of continuous functions with the support of $f_n$ contained in a ball of radius $1/n$ around $0$ and satisfying $0\leq f_n \leq 1$ on $S$ and $f_n(0)=1$ for every $n$. It is readily seen that this sequence of functions is not a Cauchy sequence in $C(S)$, while it is a Cauchy sequence converging to the zero function $0\in L^2(S,\mu)$. If $\delta_0$ were $\|\cdot\|_2$-continuous, one would have the contradiction $$0=\delta_0(0)=\delta_0\big(L^2(S,\mu)\text{-}\lim_n f_n\big)=\lim_n \delta_0(f_n)= \lim_n f_n(0)=1.$$