The following seems like a very easy question to answer, but I am not able to answer it. Here's the question:
Let $\mathcal{A}$ be the collection of all subsets in $\mathbb{R}$ that can be expressed as finite unions of half-open intervals $[a, b)$. Let $µ_{0} : A \longrightarrow [ 0, + \infty ]$ be the function such that $µ_{0}(E) = \infty$ for non-empty $E$ and $µ_{0}(\emptyset) = 0$. Show that the Hahn-Komologorv extension $\mu$, restricted to Borel measurable sets, assigns infinite measure to all Borel measurable sets.
I'm stuck how to go about solving this problem. Any suggestions would be welcome.