Folland says in his chapter on the dual of $C_0(X)$ (continuous functions on $X$ vanishing at $\infty$):
We recall that for any LCH space $X$, $C_0(X)$ is the uniform closure of $C_c(X)$, and hence if $\mu$ is a Radon measure on $X$, the functional $I(f)=\int f\ d\mu$ extends continuously to $C_0(X)$ iff it is bounded with respect to the uniform norm. In view of the equality $\mu(X)=\sup \big\{ \int f\ d\mu: f\in C_c(X), 0\le f\le 1 \big\}$ together with the fact that $|\int f\ d\mu| \le \int |f|\ d\mu$, this happens precisely when $\mu(X)<\infty$, in which case $\mu(X)$ is the operator norm of $I$.
We have therefore identified the positive bounded linear functionals on $C_0(X)$: they are given by integration against finite Radon measures.
I feel like an idiot but I just can't seem to parse this passage. Can someone link me to another explanation of this? I'm specifically having trouble understanding why the first "iff" statement is true and why the finiteness of the measure is implied exactly. I just keep reading it and it's not sinking in.
Forgive me if this is not a good question.