Problem with the existence of a iterated integral

Question

Let $$ F(x,y)= \begin{cases} 1 &\mbox{ if } x\in\mathbb{Q},\\ 0 &\mbox{ if } x\in\mathbb{R}\setminus\mathbb{Q} \end{cases} $$ Then, $\int_{0}^{1}\int_{0}^{1}F(x,y)dydx$ exists?\ I'm convinced that this integral does not exists, because if $x\in\mathbb{Q}$ then the integral $\int_{0}^{1}F(x,y)dy=1$ and 0 in the other case but my proffesor told me that it exists since there is no dependence on the variable y. I don`t know if he told me that just to confuse me or is actually true...