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.



Equivalent Definition of Measurable Function
* $1 \Rightarrow 2$ by the measurability of $\leq$.
* $2 \Rightarrow 1$ by noting that the inverse image of $\{ f(x) \leq y \}$ under the measurable function $x \mapsto (x, c)$ is $\{ f(x) \leq c \}$. The latter must be measurable, and this sufficient for measurability of $f$.


I don't know why this wouldn't be mentioned in a text book. It's mentioned here for example: <http://unapologetic.wordpress.com/2010/07/21/measurable-graphs/>


