Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mathematics and Statistics


There are many different theories of integration, several of which are not equivalent. In books such as [3] and [4] many of the theories are discussed in detail. This paper is devoted to developing an abstract theory of integration which contains many of the classical theories as special cases. In addition, the theory to be presented has been specifically designed to include the Riemann - complete integral which was recently introduced by Henstock [2]. All integrals considered will be integrals of so-called "point-interval functions." This is a modification of the idea of integration of interval functions which was first introduced by Burkill [l]. A general theory of integration of point-interval functions on [a,b] will be presented first. Then four special cases will be given, and it will be shown that no two of these integration processes are equivalent, although they are all covered by the general theory. Finally, a theory of integration over families of intervals will be introduced. Throughout this paper, R+ will be used to denote the set of all real numbers, R will be used to represent the set of all positive real numbers, RP will be used to represent Cartesian p-space, and [a,b] will be used to denote a nondegenerate closed sub-interval of R.

Library of Congress Subject Headings

Calculis, Integral



Number of Pages



South Dakota State University