Research Of Set-valued And Fuzzy-valued Measures, Integrals And Metric Spaces

This thesis is composed of three parts. Part I includes Chapter2and Chapter3. In Chapter2, a new integral of real-valued functions with respect to a set-valuedmeasure is introduced. Some of its properties and convergence theorems are obtained.In Chapter3, generalized fuzzy number measure and integral are studied. Firstly,we establish the Lebesgue type decomposition theorem and the weak Radon-Nikody′mtheorem for generalized fuzzy number measures. And then, we introduce the integral ofreal-valued functions with respect to a generalized fuzzy number measure. We discusssome of its properties and obtain convergence theorem. Part II includes Chapter4.We attempt to deal with special generalization of measure theory considering valuedin the set of gradual numbers. The concepts of gradual number-valued measures,gradual number-valued measurable functions and gradual number-valued integrals ofgradual number-valued measurable functions with respect to gradual number-valuedmeasures are introduced, their properties are investigated. Part III includes Chapter5. In this part, firstly, we introduce a new concept of probabilistic metric space,which is a generalization of Menger probabilistic metric space, and investigate sometopological properties of this space and related examples. Also, we prove some fixedpoint theorems and present an example to illustrate the main theorems. Finally, theconcept of gradual metric spaces is introduced. basic concepts with respect to topologyin the gradual metric space are presented and their properties are discussed.