New Conditions for the Egoroff Theorem in Non-additive Measure Theory

This paper gives a new necessary condition and a new sufficient condition for the Egoroff theorem in non-additive measure theory. The new necessary condition is condition (M), which is newly defined in this paper, and the new sufficient condition is the conjunction of null continuity and condition (M). The new sufficient condition is strictly weaker than both of known two sufficient conditions: continuity and the conjunction of strong order continuity and property (S). The new necessary condition is strictly stronger than the known necessary condition: strong order continuity.