  Given positive integers $\ell<n$ and a real $d\in (\ell,n)$, we construct
sets $K\subset \mathbb R^n$ with positive and finite Hausdorff $d-$measure such
that the Radon-Nikodym derivative associated to all projections on
$\ell-$dimensional planes is not an $L^p$ function, for all $p>1$.
