The Space Of Fuzzy Chquet Integrable Functions And Set-valued Fuzzy Integral

The paper has three parts.The first part is about some characteristics of the space of fuzzy Choquet integrable functions.On the basis of the space which is constructed by the fuzzy Choquet integrable and measurable nonegative functions,we give the definition of the function space L_p(p ≥ 1),then,we define the pseudo-norm || · ||_p on L_p and prove that (L_p, || · ||_p) is a pseudo-normed space.At the last part of this chapter,we study the completeness of the function space (L_p, || · ||_p) and prove that (L_p, || · ||_p) is a complete pseudo-normed space.The secend part is about the fuzzy integral with respect to set-valued fuzzy measure.Firstly,we give the definition of set-valued fuzzy measure whose valued-field is Φ₀(R₊^m) and study some characteristics of it,including null-additive(sub.), autocontinuous from above(below) and uniform-autocontinuous etc. Secondly,we introduce the notion of (pseudo-)convergence in set-valued fuzzy measure of the measurable function sequences and discuss some properties of it. Thirdly,we define the fuzzy integral with respect to set-valued fuzzy measure by the choices of it and gain the convergence theory of this kind of fuzzy integral.In the last part we first discuss about some generalized properties of the fuzzy Choquet integral of measurable set-valued functions,when we regard the whole integral as a set function,we prove that it is a set-valued fuzzy measure.In the end,we study the inherited characteristics(p.g.p and (S))of the set function whichis defined by the generalized fuzzy-valued Choquet integrals.