A new necessary and sufficient condition for the Egoroff theorem in non-additive measure theory

This paper shows that a newly defined condition, called condition (M), is a necessary and sufficient condition for the Egoroff theorem in non-additive measure theory. The existing necessary and sufficient conditions for the Egoroff theorem are described by a doubly-indexed sequence of measurable sets, while condition (M) is described by a singly-indexed sequence of measurable sets.