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A study of functions of Baire class one
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Type
Thesis
Author
Zulijanto, Atok
Supervisor
Tang, Wee Kee
Abstract
The study of functions of Baire class one, or simply Baire-1 can be traced back to Baire’s paper [Ba] in 1899. A real-valued function on a metrizable space is said to be of Baire class one if it is the pointwise limit of a sequence of continuous functions. In 2001, Lee, Tang and Zhao [LTZ] provided an equivalent definition of Baire-1 functions in terms of ε-gauge similar to that in the definition of continuity of a function. Recently, ordinal indices which measure the complexity of Baire-1 functions have been also studied by several authors. In this thesis, we study the gauges associated to Baire-1 functions and use them to study oscillation index β of Baire-1 functions.
The ε-gauge of Baire-1 functions originates from the study of Kurzweil-Henstock integrals. It was shown that the gauge for Kurzweil-Henstock integrals can always be taken to be measurable. In Chapter 2, we show that the ε-gauge δ for bounded Baire-1 functions can be chosen to be well-behaved. More precisely, for any bounded Baire-1 function f : K → R on a compact metric space K and for any ε > 0, there exists an upper semicontinuous ε-gauge δ of f with finite oscillation index (see also [ATZ]).
The subsequent chapters are devoted to the study of oscillation index β of Baire-1 functions on a Polish space via their gauges. In chapter 3, we introduce an ordinal index of the gauges associated to Baire-1 functions that we call zero index. We establish the equivalence of the ε-oscillation index of a Baire-1 function and the zero index of its corresponding ε-gauge. Some immediate applications are also provided in this chapter.
In the last two chapters, we provide further applications of the gauge approach. We compute the oscillation index of the product and quotient of Baire-1 functions on a Polish space and present the results in Chapter 4. Our result on the product sharpens Theorem 6.5 in [LT1]. Chapter 5 deals with the oscillation index of composition of a Baire-1 function with a function having at most finitely many points of discontinuity. Examples are given to show that the results are optimal.
The ε-gauge of Baire-1 functions originates from the study of Kurzweil-Henstock integrals. It was shown that the gauge for Kurzweil-Henstock integrals can always be taken to be measurable. In Chapter 2, we show that the ε-gauge δ for bounded Baire-1 functions can be chosen to be well-behaved. More precisely, for any bounded Baire-1 function f : K → R on a compact metric space K and for any ε > 0, there exists an upper semicontinuous ε-gauge δ of f with finite oscillation index (see also [ATZ]).
The subsequent chapters are devoted to the study of oscillation index β of Baire-1 functions on a Polish space via their gauges. In chapter 3, we introduce an ordinal index of the gauges associated to Baire-1 functions that we call zero index. We establish the equivalence of the ε-oscillation index of a Baire-1 function and the zero index of its corresponding ε-gauge. Some immediate applications are also provided in this chapter.
In the last two chapters, we provide further applications of the gauge approach. We compute the oscillation index of the product and quotient of Baire-1 functions on a Polish space and present the results in Chapter 4. Our result on the product sharpens Theorem 6.5 in [LT1]. Chapter 5 deals with the oscillation index of composition of a Baire-1 function with a function having at most finitely many points of discontinuity. Examples are given to show that the results are optimal.
Date Issued
2008
Call Number
QA331.5 Zul
Date Submitted
2008