PM-Fuzzy Valued Integral And Fuzzy Valued Functions Sequence

The paper mainly has two parts.The first part is about PM-fuzzy valued integral.First,we extend the integrable function of (G) fuzzy integral to all real-valued measurable functions,get some important properties and discuss the absolute of the integrabled functions in the sense of above fuzzy integral.Next,by using the pseudo-multiplication,we extend the integrable function of (G) fuzzy integral to nonnegative fuzzy valued functions which is called pseudo-multiplication derivational fuzzy valued functions (PM-fuzzy valued integral), discuss some basic properties and get Levi theorem and Fatou lemma.At the last,regarding wholly this kind of integrals as a set-valued functions whose valued-field is nonnegative fuzzy number,we study the inherited characteristics which is about original fuzzy measure.The second part is about the fuzzy valued functions sequence. One hand,we introduce the concepts of the convergence in fuzzy measure,the almost uniform convergence and the almost everywhere convergence for the sequence of fuzzy valued functions,further Egoroff theorem,Lebesgue theorem in fuzzy measure space and monotone measure space are proved.On the other hand,we give the definition of the fuzzy valued functions sequence uniform integrability and uniform boundedness for PM-fuzzy valued integral ,make the fuzzy valued measure theory perfect.