  We consider a uniformly rectifiable set $\Gamma \subset \mathbb R^n$ of
dimension $d<n-1$. By using degenerate elliptic operators on the complement
$\Omega = \mathbb R^n \setminus \Gamma$, Guy David, Svitlana Mayboroda, and the
author introduced a notion of harmonic measure on $\Gamma$.
  We prove in the present article that this harmonic measure on $\Gamma$
satisfies the $A^\infty$-property, that is the harmonic measure and the
$d$-dimension Hausdorff measure on $\Gamma$ are mutually absolutely continuous
in a quantitative and scale invariant way. Thus, we give an alternate proof of
a recent theorem of David and Mayboroda, which itself extends a result of
Hofmann and Martell to the case where the uniformly rectifiable set $\Gamma$ is
not of codimension 1.
  The proof is surprisingly simple - in particular does not follow the route
used by David and Mayboroda, or by Hofmann and Martell - but is specific to the
case when $d<n-1$.
