  Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of
codimension $1$. We show that every harmonic function is
$\varepsilon$-approximable in $L^p(\Omega)$ for every $p \in (1,\infty)$, where
$\Omega := \mathbb{R}^{n+1} \setminus E$. Together with results of many authors
this shows that pointwise, $L^\infty$ and $L^p$ type
$\varepsilon$-approximability properties of harmonic functions are all
equivalent and they characterize uniform rectifiability for codimension $1$
Ahlfors-David regular sets. Our results and techniques are generalizations of
recent works of T. Hyt\"onen and A. Ros\'en and the first author, J. M. Martell
and S. Mayboroda.
