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Motivated by the recent progress on positive mass theorem for asymptoticallyflat manifolds with arbitrary ends and the Gromov's definition of scalarcurvature lower bound for continuous metrics, we start a program on thepositive mass theorem for asymptotically flat manifolds with $C^0$ arbitraryends. In this work as the first step, we establish the positive mass theorem ofasymptotically 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 anon-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. Newtechniques are developed to deal with non-compactness of the singular set.

Singular positive mass theorem with arbitrary ends