Uniqueness of Regular Borel measures on the Baire sets

Uniqueness of Regular Borel measures on the Baire sets

I'm having problems proving the following theorem:
Theorem: If two regular Borel measures agree on all Baire sets, then they
agree on all Borel sets.
I have shown that $\mu(C) = \nu^*(C)$ for all compact sets, where
$\nu^*(C) = inf\{\nu(E) : C \subseteq E \in \mathcal{B_O}\}$,
$\mathcal{B_O}$ denotes the Baire sets, and $\nu = \mu_{|\mathcal{B_O}}$
is the Baire contraction of $\mu$. So obviously two regular Borel measures
agreeing on Baire sets, agree on the compact sets. But how do i extend
this uniqueness to ALL borel sets?
Book(Halmos) says to use this theorem;
Theorem: If $\mathcal{F}$ is the collection of all finite disjoint unions
of compact sets, then $\sigma(\mathcal{F}) = \mathcal{B}$.
(If $\mathcal{F_O}$ is the collection of all finite disjoint unions of
sets of compact $G_\delta$'s, then $\sigma(\mathcal{F_O}) =
\mathcal{B_O}$)
Where $\mathcal{B}$ = $\sigma(\mathcal{C})$ are the Borel sets.
Thanks in advance.