The autocontinuity of set function and the fuzzy integral (Q1068228)

The autocontinuity and some other concepts of a set function are introduced and Sugeno's fuzzy measure with some annexed conditions is studied. On a fuzzy measure space, Egoroff's theorem is proved. It is also proved that F-mean convergence is equivalent to convergence in measure for a sequence of measurable functions. Finally, some convergence theorems of a sequence of fuzzy integrals are proved, especially, using the concept of autocontinuity, a necessary and sufficient condition for the convergence in measure theorem is given.