Fuzzy Sets and Systems 45 (1992) 59-68 59 North-Holland On the pseudo-autocontinuity of fuzzy measures Sun Qinghe Mathematical Teaching and Research Centre, Qinghai University, Xining, Qinghai, China Received December 1989 Abstract: The property (PS) of fuzzy measures is introduced. A set of characterizations of the pseudo-autocontinuity of fuzzy measures are given. The equivalence of pseudo-autocontinuity from above and pseudo-autocontinuity from below is established. The convergence pseudo-in measure is investigated. Keywords: Measure; fuzzy measure; pseudo-autocontinuity; convergence pseudo-in measure. 1. Introduction Just as the autocontinuity [1] of fuzzy measures, the pseudo-autocontinuity of fuzzy measures is also an important structural characteristic of fuzzy measures used in the study of fuzzy measures and fuzzy integrals, first introduced by Wang [2]. As elaborated in [2], many statements from classical measure theory, e.g., Reisz's theorem and Egoroff's theorem, remain valid for fuzzy measures with pseudo-autocontinuity and there are also some interesting applications to the fuzzy integral theory. In the present paper, we study the concept in detail. In Section 2, we first recall and give some basic definitions and propositions about fuzzy measures and the pseudo-autocontinuity of fuzzy measures. In Section 3, we introduce the definition of property (PS) of fuzzy measures and show that a fuzzy measure is pseudo-autocontinuous if and only if it is pseudo-null-additive and has the property (PS). Also we give a lot of characterizations for the pseudo-autocontinuity of fuzzy measures and we establish the equivalence of pseudo-autocontinuity from above and pseudo-autocontinuity from below for a fuzzy measure. In Section 4, we study the convergence pseudo-in measure thoroughly. It will be seen that many classical results relating to the convergence in measure are still true for the convergence pseudo-in measure when fuzzy measures are pseudo-autocontinuous. 2. Basic definitions and results Let X be a nonempty set, 0%be a sigma-algebra of subsets of X, ~" 0%---~ [0, o~) be a fuzzy measure, i.e., with monotonicity and continuity and /~(tt)= 0. We shall work throughout with the fixed fuzzy measure space (X, 0%,/t). Definition 2.1. A fuzzy measure # is called pseudo-null-additive, denoted by p.0-add., if we have #(B U C) = #(C), whenever A e 0%, B c A n 0%, C e A n 0%, #(A - B) = #(A). Proposition 2.1. The following statements are equivalent: (1) /~ is p. O-add. ; (2) p(B n C) --/~(C), whenever A e 0%, B e A n 0%, C ~ A n 0%, ~(A) =/~(B); (3) p(C u (A - B)) =/~(C), whenever A ~ 0%, B ~ A n 0%, C c A o 0%, l~(A ) -- k~(B ). 0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved