Alessandro Fonda - The Kurzweil-Henstock Integral for Undergraduates: A Promenade Along the Marvelous Theory of Integration

Compact Textbooks in Mathematics

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The author with Jaroslav Kurzweil in September 2008

This book is the outcome of the beginners courses held over the past few years for my undergraduate students. The aim was to provide them with a general and sufficiently easy to grasp theory of the integral. The integral in question is indeed more general than Lebesgues in , but its construction is rather simple, since it makes use of Riemann sums which, being geometrically viewable, are easily understandable.

This approach to the theory of the integral was developed independently by Jaroslav Kurzweil and Ralph Henstock since 1957 (cf. [5, 8]). A number of books are now available [1, 4, 6, 7, 913, 1519, 21]. However, I feel that most of these monographs are addressed to an expert reader, rather than to a beginner student. This is why I wanted to maintain here the exposition at a very didactical level, trying to avoid as much as possible unnecessary technicalities.

The book is divided into three main chapters and five appendices, which I now briefly describe, mainly as a guide for the lecturer.

In the second chapter, the theory is extended to real functions of several real variables. No difficulties are encountered while considering functions defined on rectangles. When the functions are defined on more general domains, however, an obstacle arises concerning the property of additivity on subdomains. It is then necessary to limit ones attention to functions which are integrable according to Lebesgue, after having introduced the concept of measurable set. On the other hand, for the Fubini Reduction Theorem there is no need to deal with Lebesgue integrable functions. It has a rather technical but conceptually simple proof, which only makes use of the KurzweilHenstock definition. In the Theorem on the Change of Variables in the integral, once again complications may arise (see, e.g., [2]), so that I again decided to limit the discussion only to functions which are integrable according to Lebesgue. The same goes for functions which are defined on unbounded sets. These difficulties are intrinsic, not only at an expository level, and research on some of these issues is still being carried out.

The third chapter illustrates the theory of differential forms. The aim is to prove the classical theorems carrying the name of Stokes, and Poincars theorem on exact differential forms. Dimension 3 has been considered closely: indeed, the theorems by StokesCartan and Poincar are proved in this chapter only in this case, and the reader is referred to Appendix B for the general proof. Also, I opted to discuss only the theory for