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Sets of everywhere singular functions

Kechris, Alexander S. (1985) Sets of everywhere singular functions. In: Recursion theory week. Lecture notes in mathematics. No.1141. Springer-Verlag , Berlin, pp. 233-244. ISBN 978-3-540-15673-4.

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In this paper we present a simple general method for demonstrating that in certain function spaces various sets consisting of functions that exhibit at every point a prescribed kind of singularity form a coanalytic but not Borel set. We illustrate this method by providing new proofs that the set of nowhere differential continuous functions on [0,1] is (coanalytic but) not Borel and similarly for the set of continuous functions on [0,1] which fail everywhere to have a unilateral derivative (including ± ∞) -- the so called Besicovitch functions. These results were originally proved by Mauldin, [Mau] and unpublished, respectively. We also give a new example of a coanalytic not Borel set, namely the set of integrable functions with everywhere divergent Fourier series. Finally, we formulate an abstract theorem, which includes as simple instances all the above and other similar examples.

Item Type:Book Section
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Additional Information:© 1985 Springer-Verlag. Research partially supported by NSF Grant.
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Series Name:Lecture notes in mathematics
Issue or Number:1141
Record Number:CaltechAUTHORS:20130528-102414726
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Official Citation:Kechris, A. (1985). Sets of everywhere singular functions. Recursion Theory Week. H.-D. Ebbinghaus, G. Müller and G. Sacks, Springer Berlin Heidelberg. 1141: 233-244.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38685
Deposited By: Tony Diaz
Deposited On:09 Mar 2020 14:56
Last Modified:09 Nov 2021 23:39

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