  Let $A$ be a compact set in $\Rp$ of Hausdorff dimension $d$. For
$s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^{s,A}$ is the unique Borel
probability measure with support in $A$ that minimizes $$
\Is(\mu):=\iint\Rk{x}{y}{s}d\mu(y)d\mu(x)$$ over all such probability measures.
In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then
$\mu^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional
Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.
