  The positive energy theorem for weighted asymptotically flat spin manifolds
was proved by Baldauf and Ozuch \cite{BO}, and for non-spin case by Chu and Zhu
\cite{CZh}. In this paper, we generalize the positive energy theorem for
3-dimensional asymptotically anti-de Sitter initial data sets to weighted
asymptotically anti-de Sitter initial data sets assuming that the weighted
dominant energy condition holds.
