On sufficient conditions for the Egoroff theorem of an ordered topological vector space-valued non-additive measure

Abstract

In this paper, we consider an ordered vector space endowed with a locally full topology, which is called an ordered topological vector space. We show that the Egoroff theorem remains valid for the ordered topological vector space-valued non-additive measure in the following four cases. The first case is that the measure is strongly order totally continuous; the second case is that the measure is strongly order continuous and possesses an additional continuity property suggested by Sun in 1994 when the ordered topological vector space has a certain property; the third case is that the measure is continuous from above and below when the topology is locally convex; the fourth case is that the measure is uniformly autocontinuous from above, continuous from below and strongly order continuous when the topology is locally convex. Our results are applicable to several ordered topological vector spaces.