Show that $f$ is measurable and $B$ is a Borel set $\rightarrow$ $f^{-1}(B)$ is measurable.
Hint: $\{A:f^{-1}(A) \in \mathbb{M}\}$ is a $\sigma$-algebra containing the open sets. (Here $\mathbb{M}$ is the set of all measurable sets)
So $B$ is a Borel set, which means it is a set formed from countable unions, intersections and relative complements of open sets.
I'm a bit lost here, the definition of a measurable function that I'm trying to use is that $f^{-1}(U)$ is measurable for any open set $U$. Any advice is appreciated!