- The fundamental theorem of calculus for the Kurzweil-Henstock integral in Euclidean space

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# The fundamental theorem of calculus for the Kurzweil-Henstock integral in Euclidean space

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Type

Thesis

Author

Cabral, Emmanuel Abiog

Supervisor

Lee, P. Y. (Peng Yee)

Abstract

As the title suggests, the central idea in this thesis is the characterization of the primitive of a Kurzweil-Henstock integrable function.

Chapter 1 contains the preliminaries. It is a brief discussion of the known ideas, results and definitions which are going to be used in the succeeding chapters. Certain chapters like Chapters 4 and 5 also contain one section for preliminaries to provide easy recall for the reader.

Chapter 2 provides two versions of the characterization. The first one involves the integrand explicitly. The second version is a complete characterization of the primitive without explicit reference to the integrand. This description of the integral has also been written in closed form. The sections of Chapter 2 have been accepted for publication in Real Analysis Exchange.

Chapter 3 essentially gives a characterization of the McShane integral in m-dimensional Euclidean space. It is well-known that the Lebesgue and McShane integrals are equivalent. Although the characterizations of the Kurzweil-Henstock and the McShane integrals are very similar in form, there are major differences. For instance, the characterization for McShane gives a primitive of bounded variation. Another difference is that differentiation is defined differently for McShane.

Chapter 4 give a generalization of the characterizations obtained from Chapter 2 and Chapter 3. The main theorem can be considered as a fundamental theorem for division spaces. As this fundamental theorem covers both the Kurzweil-Henstock and McShane integrals as examples, other examples can also be given such as the AP integral and the Henstock-Stieltjes integral.

Chapter 5 deals with convergence theorems without pointwise convergence. The most crucial result is the extension to the m-dimensional case of Kurzweil's convergence theorem. These convergence theorems are also studied in relation to the convergence theorems found in Chapter 2.

Chapter 1 contains the preliminaries. It is a brief discussion of the known ideas, results and definitions which are going to be used in the succeeding chapters. Certain chapters like Chapters 4 and 5 also contain one section for preliminaries to provide easy recall for the reader.

Chapter 2 provides two versions of the characterization. The first one involves the integrand explicitly. The second version is a complete characterization of the primitive without explicit reference to the integrand. This description of the integral has also been written in closed form. The sections of Chapter 2 have been accepted for publication in Real Analysis Exchange.

Chapter 3 essentially gives a characterization of the McShane integral in m-dimensional Euclidean space. It is well-known that the Lebesgue and McShane integrals are equivalent. Although the characterizations of the Kurzweil-Henstock and the McShane integrals are very similar in form, there are major differences. For instance, the characterization for McShane gives a primitive of bounded variation. Another difference is that differentiation is defined differently for McShane.

Chapter 4 give a generalization of the characterizations obtained from Chapter 2 and Chapter 3. The main theorem can be considered as a fundamental theorem for division spaces. As this fundamental theorem covers both the Kurzweil-Henstock and McShane integrals as examples, other examples can also be given such as the AP integral and the Henstock-Stieltjes integral.

Chapter 5 deals with convergence theorems without pointwise convergence. The most crucial result is the extension to the m-dimensional case of Kurzweil's convergence theorem. These convergence theorems are also studied in relation to the convergence theorems found in Chapter 2.

Date Issued

2002

Call Number

QA312 Cab

Date Submitted

2002