Suppose $f: \mathbb{R} \to \mathbb{R} $ is a function. Is it equivalent that:
1) $f$ is measurable
2) the area under $f$ (i.e $\{ (x,y)\ | \ f(x)\leq y\} $) is measurable in the product measure of the Borel measure and Lebesgue measure.
We think the answer is yes, gave a proof, cannot find anything wrong with it. However, we feel strange as we cannot find the result mentioned in any textbook and it seems very basic.