by Jaroslav Kurzweil (Mathematical Institute of the Academy of Sciences, the Czech Republic)

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 book HK integration is treated only on compact one-dimensional intervals.

The set of convergent sequences of HK-integrable functions is singled out by an elementary convergence theorem. 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: 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.

• Integrable Functions and Their Primitives
• Gauges and Borel Measurability
• Convergence
• An Abstract Setting
• An Abstract Setting with D Countable
• Locally Convex Topologies Tolerant to Q-Convergence
• Topological Vector Spaces Tolerant to Q-Convergence
• P as a Complete Topological Vector Space
• Open Problems