  Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify
the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the
vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x -
y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in
\mathcal{L}(V_{s})$ if and only if the integral above has a principal value at
$a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the
precise representative of $V_{s}$ at $a$ coincides with the principal value of
the integral. We also study the existence of Lebesgue points for the Cauchy
integral of the intrinsic probability measure associated with planar Cantor
sets, which leads to challenging new questions.
