A Research On The Theories Of Fuzzy Metrics And Convergence On Generalized Measure Space

As a branch of fuzzy sets theory, the theory of generalized measure(including non-additive measure and fuzzy measure) was formed in the 1970 s. And together with and generalized integral, are the extension of classical measure and integral.It has a close connection with harmonic analysis, differential equations, difference equations and optimization, and it is widely used in multi-criteria decision making,information integration, pattern recognition and aggregation functions. Besides,fuzzy metric space theory is an important topic in fuzzy topology theory. The theory of fuzzy metric space that based on triangular norm has attracted much interest in recent ten years. This concept has been widely used in various papers which devoted to completion, convergence, compactness, uniformly continuous,fixed point and so on. In addition, cross-disciplinary researches among fuzzy metric and generalized measure, Domain, color image filtering were also given as the application of fuzzy metric.Convergence problem is the core issue in topology and measure theory. The use of metric theory in convergence problem is the important branch of topology measure, and it is also an important sign of close connection between measure theory and topology. This thesis concerned with the connection between the generalized measure theory and fuzzy metric theory. We will focus on constructing fuzzy metric on the generalized measure space, then investigate the convergence problem and the application of entension of generalized measure. The study will make the generalized measure theory and fuzzy metric theory more perfect, and provide a more solid foundation for the applications of generalized measure theory.Based on two kind of generalized measure(decomposable measure and fuzzy measure), the thesis mainly focused on three aspects as follows. Firstly, constructing the mutual generated approach for generalized measure and fuzzy metric. Secondly, discussing the correspondence properties between the generalized measure and fuzzy metric spaces. Thirdly, apply the above results to the convergence theory and extension theory of generalized measure space.The dissertation consists of four parts and six chapters. Part 1 contributed to the generalized metric on the decomposable measure space. By constructing a generalized metric on the measurable sets of a given σ-⊥-decomposable measure, we discussed several properties such as completeness and continuity of the constructed generalized metric space. Consequently, we proved that the μ-separablility and nonatom of the σ-⊥-decomposable measure can be characterized in the constructed generalized metric space.Part 2 devoted to studying the fuzzy metric on fuzzy measure space. By constructing a fuzzy metric on the fuzzy measurable sets, we studied the relations between them. Similar to Part 1, we constructed a fuzzy metric on the associated quotient sets from a given fuzzy measure. Furthermore, we studied some basic properties of the constructed fuzzy metric space such as completeness and continuity etc. In addition, we investigated the charaterization of nonatomic of the fuzzy measure and the corresponding properties of constructed fuzzy metric space.The results suggest that the standard approach for obtaining a metric from a given probability can be generalized to the fuzzy measure for the case of the t-norm min.Part 3 concerned with the application of generalized pseudo-metric to the extension of decomposable measure. We proved that the extension of non-strict Archimedean t-conorm-based σ-decomposable measure, can be formulated as from a subset of certain generalized pseudo-metric space into the closure of the subset.The extension result via generalized pseudo-metric is equivalent to the completion of t-conorm-based σ-decomposable measure and the well-known Carath′edory result, and the method proposed in this paper is intuitive and effective.Part 4 studied the Vitali-Hahn-Saks theorem on decomposable measure space.Based on Part 1, we derived the problem from the setwise convergence of decomposable measure sequences. By virtue of Baire Category theorem on the generalized metric space, we can prove the Vitali-Hahn-Saks theorem and Nikodym theorem on decomposable measure space. The result reveals that the uniform absolute continuity is a consequence of setwise convergence under certain conditions. The measure concepts can be translated into topology concepts.