Integration on Rn and manifolds which are immersed in Rn

In this thesis, integration of scaler functions on $\mathbb{R}^n$and on manifolds which are immersed in $\mathbb{R}^n$ is examined indetail. For this purpose; some definition, propositions and theorems-which are needed for the remaining sections- are presented in thefirst section of total eight. In the second, the concepts of measurezero and content zero are introduced. Furthermore examples ofmeasure zero and content zero sets are provided and interrelationbetween these two concepts are expressed and some basic results areproved. In the third section, necessary and sufficient condition forintegrability of functions are given, and some general properties ofintegration are expressed. Moreover the integration on a closedrectangle in $\mathbb{R}^n$ is generalized to on an arbitrarybounded subset of $\mathbb{R}^n$ and again; some basic properties ofit are investigated. In the fourth part, Fubini's Theorem is statedand proved. In the fifth part, Improper Integrals are defined andgeneral properties of them are presented. Further, interrelationbetween the ordinary and improper integral are expressed. In thesixth part, Partition of Unity is defined and its existence isproved. In the seventh part, Change of Variables Theorem isexpressed. Finally, in the last part, manifolds in $\R^n$ aredefined and various examples are given. Moreover, the concept ofintegration of scaler functions is generalized to on such manifoldsand some of its properties are investigated.