Prove that $f$ is measurable

Question

Let $f$ be a real valued application on a measurable set $(X,A)$ show that if the set $\{{x \in X : f(x) > r}\}$ is measurable for every $r \in \mathbb Q$, then $f$ is measurable

Answer 1

As I said in the comments, first observe that intervals with boundary points in $\mathbf{Q}$ generate the Borel $\sigma$-field of $\mathbf{R}$. Now use an easy but very useful lemma:

$f:(X,\mathcal{A})\rightarrow (Y,\mathcal{B})$ a function. If $\mathcal{C}$ is a generating set of $\mathcal{B}$ , then $f^{-1}(\sigma(\mathcal{C}))=\sigma(f^{-1}(\mathcal{C}))$

I think one can prove this using "Good Set Principle". Now we need to show $f^{-1}(\sigma(\mathcal{C}))\subset \mathcal{A} \Leftrightarrow \sigma(f^{-1}(\mathcal{C}))\subset \mathcal{A} $ . The question says, $f^{-1}(\mathcal{C})\subset \mathcal{A} $, as $\mathcal{A}$ is a $\sigma$-field, this says, $\sigma(f^{-1}(\mathcal{C}))\subset \mathcal{A} $. So, we are done.