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Theory of nonabsolute integration
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Type
Thesis
Abstract
The main objective of this thesis is to define a nonabsolute integral measure theoretically. More precisely we define an integral of the Henstock type, called the H-integral, on measure spaces with a locally compact Hausdorff topology that is compatible with the measure. Relevant results pertaining to the H-integral are established.
In Chapter 1, we define the H-integral and derive the properties that are fundamental to an integral. We describe in Section 1.1 how certain objects in the space are chosen to be generalised intervals and relate the definition to some concrete examples. The H-integral is defined in Section 1.2 and we prove that it includes the well-known Kurzweil-Henstock integral [18] on the real line. The basic properties that hold true for the H-integral, in particular, the Henstock's lemma and the monotone convergence theorem, are derived in Section 1.3.
Chapter 2 aims to relate the H-integral to known integrals. In Section 2.1, we define the M-integral, which is a McShane-type integral, and prove that a function is M-integrable if and only if it is absolutely H-integrable. The domains of H-integration and M-integration are also extended to measurable sets. Subsequently in Section 2.2 we establish the equivalence between the M-integral and the Lebesgue integral. we also show that a function H-integrable on an elementary set is Lebesgue integrable on a portion of the elementary set. In Section 2.3 we establish the fact that the H-integral includes the Davies as well as the Davies-McShane integral defined by Henstock in [13]. This is done by establishing the equivalence between the Lebesgue integral and the Davies as well as the Davies-McShane integral. The conclusion here is that if a function is measurable then the absolute H-integral, the M-integral, the Lebesgue integral, the Davies integral and the Davies-McShane integral and the Davies-McShane integral are all equivalent.
Further results of the H-integral are given in Chapter 3. We begin by proving in section 3.1 that H-integrable functions are measurable and proceed to give a necessary and sufficient condition for a function to be H-integrable. We also prove that the H-integral is genuinely a nonabsolute one by constructing an example which is H-integrable but not absolutely H-integrable. Two concepts very relevant to the H-integrability, namely the generalised absolute continuity and equi-integrability, are introduced in Section 3.2 and some results involving these concepts are proved. Section 3.3 is devoted to proving the convergence theorems of the H-integral. We start with the proofs of the equi-integrability theorem and the basic convergence theorem and illustrate how the mean convergence theorem can be proved with the aid of the two former theorems. The controlled convergence theorem is proved in s few lemmas and by applying the basic convergence theorem.
Chapter 4 is the most important part of this thesis. We generalise our work in [23] for the H-integral in this chapter. The main theorem, namely the Radon -Nikodym theorem for the H-integral, is proved in Section 4.1 with which we give a descriptive definition of the H-integral in Section 4.2. By imposing a different condition, a second version of the main theorem and subsequently a second descriptive definition of the H-integral are also given. The purpose of Section 4.3 is to report on our findings in [23]. some results corresponding to those we prove in Section 4.1 are given for the Euclidean space setting. We also show how some known results on the real line, for example, the fundamental theorem of calculus for the Kurzweil-Henstock integral, can be deduced.
In Chapter 1, we define the H-integral and derive the properties that are fundamental to an integral. We describe in Section 1.1 how certain objects in the space are chosen to be generalised intervals and relate the definition to some concrete examples. The H-integral is defined in Section 1.2 and we prove that it includes the well-known Kurzweil-Henstock integral [18] on the real line. The basic properties that hold true for the H-integral, in particular, the Henstock's lemma and the monotone convergence theorem, are derived in Section 1.3.
Chapter 2 aims to relate the H-integral to known integrals. In Section 2.1, we define the M-integral, which is a McShane-type integral, and prove that a function is M-integrable if and only if it is absolutely H-integrable. The domains of H-integration and M-integration are also extended to measurable sets. Subsequently in Section 2.2 we establish the equivalence between the M-integral and the Lebesgue integral. we also show that a function H-integrable on an elementary set is Lebesgue integrable on a portion of the elementary set. In Section 2.3 we establish the fact that the H-integral includes the Davies as well as the Davies-McShane integral defined by Henstock in [13]. This is done by establishing the equivalence between the Lebesgue integral and the Davies as well as the Davies-McShane integral. The conclusion here is that if a function is measurable then the absolute H-integral, the M-integral, the Lebesgue integral, the Davies integral and the Davies-McShane integral and the Davies-McShane integral are all equivalent.
Further results of the H-integral are given in Chapter 3. We begin by proving in section 3.1 that H-integrable functions are measurable and proceed to give a necessary and sufficient condition for a function to be H-integrable. We also prove that the H-integral is genuinely a nonabsolute one by constructing an example which is H-integrable but not absolutely H-integrable. Two concepts very relevant to the H-integrability, namely the generalised absolute continuity and equi-integrability, are introduced in Section 3.2 and some results involving these concepts are proved. Section 3.3 is devoted to proving the convergence theorems of the H-integral. We start with the proofs of the equi-integrability theorem and the basic convergence theorem and illustrate how the mean convergence theorem can be proved with the aid of the two former theorems. The controlled convergence theorem is proved in s few lemmas and by applying the basic convergence theorem.
Chapter 4 is the most important part of this thesis. We generalise our work in [23] for the H-integral in this chapter. The main theorem, namely the Radon -Nikodym theorem for the H-integral, is proved in Section 4.1 with which we give a descriptive definition of the H-integral in Section 4.2. By imposing a different condition, a second version of the main theorem and subsequently a second descriptive definition of the H-integral are also given. The purpose of Section 4.3 is to report on our findings in [23]. some results corresponding to those we prove in Section 4.1 are given for the Euclidean space setting. We also show how some known results on the real line, for example, the fundamental theorem of calculus for the Kurzweil-Henstock integral, can be deduced.
Date Issued
1997
Call Number
QA303 Ng
Date Submitted
1997