Some Topics in Descriptive Set Theory and Analysis

Author: Ramsamujh, Taje Indrallal

Year: 1986

Degree: Dissertation (Ph.D.)

Advisor: Kechris, Alexander S.

Committee Members: Kechris, Alexander S.; Woodin, W. Hugh; Wolff, Thomas H.; Luxemburg, W. A. J.; De Prima, Charles R.

Option: Mathematics

DOI: 10.7907/8pdn-xf41

Abstract

Coanalytic subsets of some well known Polish spaces are investigated. A natural norm (rank function) on each subset is defined and studied by using well-founded trees and transfinite induction as the main tools. The norm provides a natural measure of the complexity of the elements in each subset. It also provides a "Rank Argument" of the non-Borelness of the subset.

The work is divided into four chapters. In Chapter 1 nowhere differentiable continuous functions and Besicovitch functions are studied. Chapter 2 deals with functions with everywhere divergent Fourier series, and everywhere divergent trigonometric series with coefficients that tend to zero. Compact Jordan sets (i.e., sets without cavities) and compact simply-connected sets in the plane are investigated in Chapter 3. Chapter 4 is a miscellany of results extending earlier work of M. Ajtai, A. Kechris and H. Woodin on differentiable functions and continuous functions with everywhere convergent Fourier series.

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