  We show that the positive mass theorem holds for continuous Riemannian
metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of
dimension less than or equal to $7$ or spin-manifolds of any dimension. More
generally, we give a (negative) lower bound on the ADM mass of metrics for
which the scalar curvature fails to be non-negative, where the negative part
has compact support and sufficiently small $L^{n/2}$ norm. We show that a
Riemannian metric in $W^{2, p}_{loc}$ for some $p > \frac{n}{2}$ with
non-negative scalar curvature in the distributional sense can be approximated
locally uniformly by smooth metrics with non-negative scalar curvature. For
continuous metrics in $W^{2, n/2}_{loc}$, there exist smooth approximating
metrics with non-negative scalar curvature that converge in $L^p_{loc}$ for all
$p < \infty$.
