Let $\Omega \subset \mathbb{R}^n$ be open. Let $1 \leq p < \infty$. I want to prove that the space of bounded functions $f:\Omega \rightarrow \mathbb{R}$ such that $\dfrac{\partial f}{\partial x_i}$ (in the weak sense) is in $L_p(\Omega)$ for every $i = 1, \ldots, n$ and such that $m_n({\rm supp}~ f) < \infty$ ($m_n$ the Lebesgue measure in $\mathbb{R}^n$) is dense in the Sobolev space $W^{1,p}(\Omega)$.
Any hint will be of great value!