Research Of Set-valued And Fuzzy-valued Choquet-Pettis Integrals Theory

Classical measure theory is a metric geometry which is used to measure the length,area or volume of matter in the objective world.The establishment of classical measure and integral theory plays a very important role in the development of many branches of mathematics.However,because of the fuzziness and non-additivity of objective things,people often meet some problems that cannot be explained by classical measure and integral theory.Thus the measure and integral theory based on fuzzy mathematics is produced,which is widely used in the research of process control,fuzzy optimization,decision theory,finance and economy and other problems.In this paper,we also devote ourselves to this work,and we study the set-valued and fuzzy-valued Choquet-Pettis integral theory.The main research content is divided into the following two parts:(1)We study the concept of set-valued Choquet-Pettis integral given by Park,and discuss the properties of set-valued Choquet-Pettis integral Park didn't introduce and study.at the same time,we also give the convergence theorem for it.(2)Based on the set-value Choquet-Pettis integral of Park,we further extend it to the fuzzyvalued measurable functions.Then we introduce a fuzzy integral of a new fuzzy-valued functions with respect to the non-additive measure,and discuss its properties of linearity,monotony,asymmetry and regularity.we also give the convergence in distribution and other convergence theorem of it.