Publication: A study of functions of Baire class one
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.<br><br>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]).<br><br>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.<br><br>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.