  Like the classical potential theory, it was conjectured that there exists
equivalence between locally and globally pluripolar and complete pluripolar
sets, namely, Problem I of Lelong, and was solved by Josefson, Bedford - Taylor
and Col\c{t}oiu. In this article, we consider complements of complete K\"ahler
domains as the generalization of closed complete pluripolar sets and prove that
there exists an equivalence between local and global existence of these sets.
