Although I understand the proof of the theorem, but I wonder about why proof of $(a)$ needs such a complex progress. The key idea is that if we can show $\mu(V - E) < \epsilon$ for any $E$, then by a complement argument we can prove $(a)$. But showing $\mu(V-E) < \epsilon $ is just a outer regularity of $\mu$, which is proved in the Riesz representation theorem, a theorem 2.14 in the same book.
Is there something I misunderstood?
Thanks!