Is a positive regular Borel measure on a compact Hausdorff space always finite?

Question

Let $\Omega$ be a compact Hausdorff space and $\mu$ a positive regular Borel measure on $\Omega$. On page 67 of the book 'C*-algebras and operator theory (G.J. Murphy)' I encounter the following statement:

"… therefore, there is a Borel set $S$ of $\Omega$ such that $\mu(S)>0$ and … for all $\omega\in S$. Since $\mu$ is regular, $$\mu(S)=\sup\{\mu(K):K \ \text{is compact and} \ K\subset S\},$$ so we may suppose that $S$ is compact. Then $\mu(S)<\infty$, again by regularity of $\mu$."

My questions about this statement:

1. Why may we suppose that $S$ is compact?
2. Why do we then have $\mu(S)<\infty$?
3. Arguing similarly for the set $S:=\Omega \ (=\text{compact}$), why can't we conclude that $\mu(\Omega)<\infty$? Moreover, since $\mu(S)\leq\mu(\Omega)$, wouldn't this automatically imply that $\mu(S)<\infty$?
4. See title.

Any suggestions are greatly appreciated!