  We prove the positive mass theorem for manifolds with distributional
curvature which have been studied in \cite{Lee2015} without spin condition. In
our case, the manifold $M$ has asymptotically flat metric $g\in C^0\bigcap
W^{1,p}_{-q}$, $p>n$, $q>\frac{n-2}{2}$. We show that the generalized ADM mass
$m_{ADM}(M,g)$ is non-negative as long as $q=n-2$, and $g$ has non-negative
distributional scalar curvature, bounded curvature in the Alexandrov sense with
its distributional Ricci curvature belonging to certain weighted Lebesgue space
and some extra conditions.
