Generalizations of Riemann Integration
Chapter 10 gives an overall view to Riemann type of integration. Riemann-Stieltjes integral and its properties are discussed. Helli’s theorems are proved and they are used for proving a representation of linear functionals in the space of continuous functions. Kurzweil-Henstock integral is defined as a most general definition for integral of functions of single variable. It is shown that this integral covers proper and improper Riemann integrals. Fundamental theorem of calculus for Kurzweil-Henstock integral is discussed. Lebesgue integral is introduced as a particular Kurzweil-Henstock integral. Relationship of basic spaces and classes of functions is presented.