| In this thesis,we introduce the concept of ◇α-measurability and establish the combined measure theory on time scales.Meanwhile,we build the relationships between Lebesgue ◇α-measurable set and Lebesgue measurable set,then the notions of ◇α-measurable function and their extended function are introduced.Moreover,some main properties of Lebesgue ◇α-measurable function are studied.Particularly,▽-measure theory and △-measure theory on time scales can be obtained by lettingα=1 and α=0,respectively.Furthermore,some criteria for ◇α-measurability of a set are derived.Based on the notion of Lebesgue ◇α-measurable functions,the Lebesgue ◇α-integral and Riemann ◇α-integral are studied and the relation between them is discussed.In addition,the concepts of Lebesgue-Stieltjes ◇α-measure,◇α-measurable function and ◇α-integral are introduced.Based on the theory of combined measurability on time scales,some basic theorems includ ing the extension theorem and the composition theorem are established.In particular,through switching the coefficient of the combined theory on time scales,we can obtain the Lebesgue-Stieltjes △-measure and Lebesgue-Sticltjes ▽-measure by taking α=1 and α=0,respectively.Additionally,several examples are provided to demonstrate the effectiveness of the obtained results in each section.Besides,according to the fuzzy theory on time scales,we introduce the concept of the directional derivative of fuzzy functions on time scales and obtain some of their basic properties. |
