Integration between the Lebesgue integral and the Henstock-Kurzweil integral : its relation to local convex vector spaces

Description

Henstock-Kurzweil (HK) integration, which is based on integral sums, can be obtained by an inconspicuous change in the definition of Riemann integration. It is an extension of Lebesgue integration and there exists an HK-integrable function f such that its absolute value [f] is not HK-integrable. In this text HK integration is treated only on compact one-dimensional intervals. The concept of convergent sequences is transferred to the set P of primitives of HK-integrable functions; these convergent sequences of functions from P are called E-convergent. The main results are: there exists a topology U on P such that (1) (P,U) is a topological vector space, (2) (P,U) is complete, and (3) every E-convergent sequence is convergent in (P,U). On the other hand, there is no topology U fulfilling (2),(3) and (P,U) being a locally convex space Contents: Basic Concepts and Properties of y-Integration; Convergence; Convergence and Locally Convex Spaces; An Auxiliary Locally Convex Space; L-Integration; M-Integration; Noncompleteness; S-Integration; R-Integration; An Extension of the Concept of y-Integration; Differentiation and Integration