Let $f:E \to \Bbb R$ be measurable. Let $B \in \operatorname{Bor}(\Bbb R)$ be a Borel set. Show that $f^{-1}(B)$ is measurable.
Solution 1:
Your proof will depend on your definition of "measurable function".
Let's say "measurable function" means $f^{-1}(G)$ is measurable for all open sets $G \subseteq \mathbb R$.
Then define $\mathcal{A} = \{A \subseteq \mathbb R \mid f^{-1}(A) \text{ measurable} \}$. Show that $\mathcal{A}$ contains all open sets and that $\mathcal A$ is a sigma-algebra. Conclude that $\mathcal A \supseteq \operatorname{Bor}(\Bbb R)$.
Note: you cannot prove $\mathcal A = \operatorname{Bor}(\Bbb R)$.