Integration between the Lebesgue Integral and the Henstock-Kurzweil Integral: Its Relation to Local Convex Vector Spaces
The main topic of this book is integration of Riemann type on a compact interval on the real line with respect to various integration bases. This concept covers many types of integration, including the Lebesgue integration, the Denjoy integration in the restricted sense, the integration introduced by Pfeffer and Bongiorno, and many others. After the general theory of integration with respect to various integration bases in Chapters 1 and 2, two chapters present a natural locally convex topology on the space of sequences of primitive functions. The problem of completeness and other topics related to this topology are discussed in detail in Chapters 5-9: particular attention is devoted to the Lebesgue integral. The last part of the book introduces a general type of differentiation, again with respect to various integration bases, and clarifies the relation between integration and differentiation.
The book is self-contained, and will be of interest to specialists in the field of real functions. It can also be read by students, since only the basics of mathematical analysis and vector spaces are required.