Nonabsolute Integration on Measure Spaces

The following sections are included:

• Dedication
• Foreword
• Preface
• Synopsis
• Contents

Ralph Henstock developed a general theory of the integral [12; 14] which includes as special cases the Denjoy–Perron integral [23], the approximate Perron integral [4], the Weiner integral [45] and the Feynman integral [34], as well as the general Denjoy integral [42]. Here, we shall consider a special case of his theory. More precisely, we shall define a Henstock-type integral on measure spaces endowed with locally compact Hausdorff topologies, and show that in cases where the topologies are metrizable, the integral defined is a nonabsolute one.

It is well known that on the real line, the absolute Henstock–Kurzweil, the McShane and the Lebesgue integrals are equivalent [21, page 110]. This result has also been generalised to higher dimensional Euclidean spaces. In this chapter, we shall establish the relationship of the H-integral to some McShane-type integrals which parallels the Henstock theory of integration on the real line. Similar results for the H-integral on metric spaces have been proved by the author in [35].

The main objective of this chapter is to establish further results of the H-integral. In Section 3.1, we shall introduce the (LG)-condition [31] and provide a necessary and sufficient condition for H-integrability in terms of this condition. We shall also show that H-integrable functions are measurable and define explicitly a function which is H-integrable but whose absolute value is not to show that the H-integral is indeed a nonabsolute one.

The Radon–Nikodým theorem is an important result which is fundamental in measure theory and is usually applied to deal with problems in abstract analysis. It also provides us with the notion of “derivative” of a function in measure spaces where the usual definition of differentiation does not apply. The main theorem we shall prove in this chapter is thus the Radon–Nikodým theorem for the H-integral. What we shall present here is a generalisation of the results we have proved in [28] which are for higher dimensional Euclidean spaces, and those in [36] which are for metric spaces. A generalisation of the Henstock–Kurzweil integral to higher dimensions is the HK-integral (see [21, Chapter 5]) which we shall define in Section 4.3.

In [24], Lee extends the Harnack extension, an important property of the Henstock–Kurzweil integral on the real line, to the HK integral in the Euclidean space and recovers the proof by means of category argument. Note that in the classical theory of the Denjoy–Perron integral, proving convergence theorems by means of the category argument in which a key step is the use of Harnack extension is a standard approach (for reference, see [21, page 47] or [41, page 253]).

The following sections are included:

• Bibliography
• Glossary
• Index