real analysis - If $f$ measurable and $B$ a Borel set, then $f^{-1}(B)$ measurable

Let $f$ be a Lebesgue measurable function and $B$ be a Borel set. Show that $f^{-1}(B)$ is also measurable.

Attempt at the proof:

Suppose $f$ is Lebesgue measurable. Then $f^{-1}((\alpha,\infty))$ is measurable as well (by definition of a measurable function), $\forall\alpha\in\mathbb{R}$. We note that $(\alpha,\infty)\in\mathcal{B}$, where $\mathcal{B}$ is the Borel $\sigma$-algebra, since:

• $\emptyset\in (\alpha,\infty)$


• $(\alpha,\infty)^c=\mathbb{R}\setminus(\alpha,\infty)\in\mathcal{B}$


• the infinite union of open sets is once again an open set

So, $(\alpha,\infty)$ is a Borel set. So let $B=(\alpha,\infty)$. Since $f$ is measurable, $f^{-1}((\alpha,\infty))$ is measurable and so $f^{-1}(B)$ is measurable.

However I'm doubtful that this is correct, since I didn't choose an arbitrary Borel set $B$. Can anyone nudge me in the right direction? Thanks.

Answer

$\{A| f^{-1}(A) \text{is Borel}\}$ is a $\sigma$-algebra containing all open intervals. What can you say now?