real-valued measurable function under a transformation.

Question

Here's the question: "Let $f \colon \mathbb{R} \to \mathbb{R}$ be measurable. Show that $f(ax)$ is measurable for all real $a$."

I know we can to look at sets of the form $\{f \geq \alpha\}$, where $\alpha$ is any real number. Yet, I'm not sure what this information gives us about the function $f(ax)$. Maybe I'm missing something obvious? Any hints are very appreciated!

Answer 1

A function $f: (X,\mathcal{A}) \to (Y,\mathcal{B})$ is measurable if $f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$. Since the function $x \mapsto ax$ is continuous, what can you say about $f^{-1}(B)$?