*Submitted Paper*

**Inserted:** 24 jun 2009

**Year:** 2009

**Abstract:**

We give a simple proof of a statement extending this Fu's result: If $\Omega$ is a set of locally finite perimeter in $*R*^2$ then there is no function $f$ in $C^1(*R*^2)$ such that $Df (x_1,x_2)=(x_2,0)$ at a.e. $(x_1,x_2)$ in $\Omega$. We also prove that every measurable set can be approximated arbitrarily closely in $L^1$ by subsets which do not contain enhanced density points.

**Keywords:**
Finite perimeter sets, enhanced density sets