Some Topics in Descriptive Set Theory and Analysis

Citation

Ramsamujh, Taje Indrallal (1986) Some Topics in Descriptive Set Theory and Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8pdn-xf41. https://resolver.caltech.edu/CaltechTHESIS:01202017-145315777

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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