I've been dealing with the Riesz Representation Theorem for measures and it is obvious that having a measure $\mu$ I can get a continuous linear functional $\mu^*$ in $C(X)^*$ where $X$ is a compact metric space.
What about the converse? That is Given $\mu^*$ in $C(X)^*$ can we fine a measure representing $\mu^*$?
I know that for each element $\mu^*$ in $C(X)^*$ such that $\mu^*(1)=1$ and $\mu^*(f)\geq 0$, when $f\geq 0$ there exists a unique measure n $X$.
Intuitively, given $\mu^* \in C(X)^*$ we can define the measure $\mu$ of a (measurable) set $S$ as $\mu^*(1_{S})$ where $1_S$ is the indicator function of $S$.
Is this telling me that the indicator function can be approximated by continuous functions?