  Motivated by the recent progress on positive mass theorem for asymptotically
flat manifolds with arbitrary ends and the Gromov's definition of scalar
curvature lower bound for continuous metrics, we start a program on the
positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary
ends. In this work as the first step, we establish the positive mass theorem of
asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is
$W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a
non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. New
techniques are developed to deal with non-compactness of the singular set.
