  We prove a structure theorem for any $n$-rectifiable set $E\subset
\mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR
condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure.
Namely, that $H^n$-almost all of $E$ can be covered by a countable union of
boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus
E$. As a consequence, for harmonic measure in the complement of such a set $E$,
we establish a non-degeneracy condition which amounts to saying that $H^n|_E$
is "absolutely continuous" with respect to harmonic measure in the sense that
any Borel subset of $E$ with strictly positive $H^n$ measure has strictly
positive harmonic measure in some connected component of
$\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing
that our result for harmonic measure is optimal. Moreover, we show that if, in
addition, a set $E$ as above is the boundary of a connected domain $\Omega
\subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness
condition, then $H^n|_{\partial\Omega}$ is absolutely continuous (in the usual
sense) with respect to harmonic measure for $\Omega$. Local versions of these
results are also proved: if just some piece of the boundary is $n$-rectifiable
then we get the corresponding absolute continuity on that piece. As a
consequence of this and recent results by
Azzam-Hofmann-Martell-Mayboroda-Mourgoglou-Tolsa-Volberg, we can decompose the
boundary of any open connected set satisfying the previous conditions in two
disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is
absolutely continuous with respect to harmonic measure and another purely
$n$-unrectifiable piece having vanishing harmonic measure.
