This is an exercise from Lieb, Loss. I need to show that no characteristic function of a set $A\subset\mathbb{R}^n$ with a finite positive measure is in $H^1(\mathbb{R}^n)$ or even in $H^{\frac{1}{2}}(\mathbb{R}^n)$. I know that that the distributional partial derivatives of such a function must be zero.
But I don't see the contradiction since in that case the derivatives are in ${\rm L}^2(\mathbb{R}^n)$.
I would also like some help with the second part. According to the book, a function is in $H^{\frac{1}{2}}(\mathbb{R}^n)$ if and only if $\int\limits_{\mathbb{R}^n}(1+2\pi|k|)|\hat{f}(k)|^2{\rm d}k$ is finite. But for an arbitrary $A$, how can I know enough about $\chi_A$ to show that this integral isn't finite?