HENSTOCK INTEGRATION ON EUCLIDEAN SPACES

Abstract

The Henstock integral is designed to integrate highly oscillatory functions which the Lebesgue integral fails to do so. It is known that the multidimensional Henstock integral includes the Lebesgue integral. The main purpose of this thesis is to provide some recent development in the m--dimensional Henstock integral. This thesis falls into four chapters, with three appendices. In Chapter One, we study the one-dimensional Henstock integral. Most of our results are real-line independent, which will help us to understand their multi-dimensional analogue in the later chapters. In Chapter Two, we introduce the m-dimensional Henstock integral. In Section 2.1, some basic properties of the integral, up to Henstock’s Lemma, are given. From Henstock’s Lemma, a new uniform Henstock’s lemma (Theorem 2.1.14) is deduced. In Section 2.2, we prove an important Theorem 2.2.15, which enables us to deduce a result of Kurzweil and Jarnik (see [20 Theorem 2.10] or Theorem 2.2.17). In Section 2.3, the uniform Henstock’s lemma leads to a further improvement of the above Kurzweil-Jarnik’s result. In Chapter Three, we give some necessary and sufficient conditions for Hnestock integrability. This is possible due the above Kurzweil-Jarnik’s result. Some multidimensional Cauchy extension results (see Corollaries 3.210 - 3.2.11) are obtained as corollaries, which will in turn play an important role in obtaining some of the main results in Chapter Four. It is also shown that for t > 0, the ? function in the definition of the Henstock integral can be taken to be measurable In the last section of this chapter, some other applications are also included. In Chapter Four, we first prove some integration by parts formulae for multidimensional Henstock integral. This is possible due to the uniform Henstock’s Lemma. Denoting the space of all Henstock integrable function on E by H(E), an integral representation theorem for the conjugate space of (H(E),

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) is obtained from the above integration by parts formulae. Moreover, this integral representation theorem enables us to characterise the multipliers for H(E). Furthermore, the proof can be modified to characterise the multipliers for CL(E), namely the space of all GAuchy-Lebesgue integrable functions on E. In the last section of Chapter Four, we prove a Banach-Steinhaus theorem for H(E) and CL(E). Three appendices, which are mainly used in Chapter Four, are also included for easy reference. The first appendice deals with functions of strongly bounded variation on E, which enables us to formulate and prove some integration by parts formula for the Henstock integral. In the second appendix, a multidimensional Riemann-Stieltjes integral is given. This Stieltjes type integral is used in the last appendix to prove a Riesz representation theorem for the conjugate space of C0(E).