Ordered Set-valued Fuzzy Measure And Measurable Functions Depicted By Ordered Set-valued Fuzzy Measure

The paper mainly has two parts.The first part:With the background on the fuzzy measure and set-valued measure, first, having introduced a new order describing sets by the definition of norms of a set on the class of subsets of an m-dimensional positive Euclid space, the general properties of this space about this order are discussed. Second, by this order a ordered set-valued fuzzy measure and some concepts of autocontinuity of set-valued mapping arc presented, then several implicit relations of the ordered set-valued fuzzy measure about autocontinuity are stud-ied. Third, on the ordered set-valued fuzzy measure space of semi-continuity from above. further investigating the regularity of a ordered set-valued fuzzy measure, the theorem of regularity depicting this ordered set-valued fuzzy measure is obtained.The second part:To begin with, on the basis of the new order, discussing the issues among which (pseudo-) almost everywhere convergence and (pseudo-) almost uniform convergence in ordered set-valued fuzzy measure of the sequence of measurable functions, Egoroff the-orems and their corresponding reverse theorems are proved. Moreover, on the ordered set-valued fuzzy measure space of semi-continuity from above, with simple function's ap-proximating, in the sense of this order Lusin theorem is achieved to show the connection of the measurability and continuity of a function. Finally, on the ordered set-valued fuzzy measure space, having adopted selection of measure to define a generalized fuzzy inte-gral,the basic properties of this integral are shown.