I'm going through the proof of Riesz Representation Theorem for $C_0(X)$, namely Theorem $6.19$ in Rudin's Real and Complex Analysis. I have the following questions from the theorem's proof, which assume the existence of a positive linear functional $\Lambda$ on $C_c(X)$ such that $|\Phi(f)| \le \Lambda(|f|) \le \|f\|$ for $f\in C_c(X)$ (see Equation 4). This is proved later, in what follows Equation 8. For reference, I've attached the proof at the end of this post. I've highlighted in blue
- Why can $\{f_n\}$ be chosen so that $$\int_X |\bar h - f_n|\, d |\mu| \xrightarrow{n\to\infty } 0$$
- Why is $|\Lambda f|\le 1$ when $\|f\| \le 1$? The inequality right before says $|\Lambda |f|| \le \|f\|$, for $f\in C_c(X)$.
- Why is each side of $(6)$ a continuous functional on $C_0(X)$? $(6)$ holds for $C_c(X)$, which is a subset of $C_0(X)$.
- How does $(6)$ imply $(7)$, i.e. $$\int_X |g|\,d\lambda \ge \sup\{|\Phi(f)| : f\in C_0(X), \|f\|\le 1\} = 1$$
- Why are $\lambda(X) \le 1$ and $|g|\le 1$ compatible only when $\lambda(X) = 1$ and $|g| = 1$ a.e. $[\lambda]$?
After this, they construct a positive linear functional $\Lambda$ that satisfies $(4)$, hence completing the proof.
If you need any clarifications, please let me know (including statements of any other theorems which have been mentioned in the above discourse, in case you do not have the book handy and want to have a look.) Thanks for any input!