I'm working on the following problem:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Borel measurable. Let $\nu(E) = \int_Ef\,d\mu$. Show that $\nu$ is a Borel measure.
Showing this is a measure is easy enough - just check what happens to nulls sets and disjoint unions. How do I know it's a Borel measure - is it because $f$ is not defined otherwise? (The problem statement doesn't say what $\mu$ is - I assume Lebesgue measure)