  Let $d \geq 2$ and $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon
measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial
projections $\pi_{x\sharp}\mu$ of $\mu$ are absolutely continuous with respect
to $\mathcal{H}^{d - 1}$ for every centre $x \in \mathbb{R}^{d} \setminus
\operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d -
1) - s$. This is sharp. In fact, for $x$ outside an exceptional set as above,
$\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$.
