I have been studying general measure theory from Mattila's Geometry of Sets and Measures in Euclidean spaces. I am stuck on proving the Borel regularity in exercise 1. The question is as follows:
Let $\nu$ be a countably additive non-negative set function on a $\sigma$-algebra, $\mathcal{A}$ of subsets of $X$. Show that it can be extended to a (outer) measure, $\nu^{*}$ on $X$ defined as: $$\nu^{*}(A) = \inf \{ \nu(B) : A \subset B, B \in \mathcal{A} \}$$ Moreover, if $\mathcal{A}$ is contained in the Borel $\sigma$-algebra, then $\nu^{*}$ is a Borel regular measure.
I have proved that it is a (outer) measure on $X$. I am struggling to prove that it is Borel regular. In particular, I am unable to show it is even a Borel measure. My guess is to use Theorem 1.7 in the book.