Order continuous of monotone set function and convergence of measurable functions sequence

In this paper, we introduce the concept of strongly order continuity of a set function, and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space. Several equivalent conditions of strongly order continuity are presented. It is shown that a necessary and sufficient condition that Lebesgue's theorem remains valid for a monotone set function is that the set function possesses strongly order continuity. (C) 2002 Elsevier Science Inc. All rights reserved.