I am trying to show that for any Lebesgue measurable set of finite positive measure $E$, the characteristic function $\chi\_E$ is not in $H^{\frac{1}{2}}(\mathbb{R}^n)$. I found somewhere that it would be enough to show instead that

$$ \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{\vert \chi\_E(x)-\chi\_E(y) \vert^2}{\Vert x-y \Vert^{n+1}} dx dy $$

is infinite. I think that the numerator is simply the sum

$$ \chi\_{E\times E^c}(x,y)+\chi\_{E^c\times E}(x,y) $$
which simplifies the problem to showing that

$$ \int\_{E} \int\_{E^c} \frac{1}{\Vert x-y \Vert^{n+1}} dx dy + \int\_{E^c} \int\_{E} \frac{1}{\Vert x-y \Vert^{n+1}} dx dy $$

is infinite, and using Fubini, I think it is enough to show that the first term is infinite. However I am having trouble trying to simplify it further, and think that I should eventually use an integral of the form $\int\_1^\infty \frac{1}{r^p}dr$ somehow.

I would appreciate any hints or helpful remarks, including those telling me that this attempt is inherently flawed.