Descriptive Set Theory and the Structure of Sets of Uniqueness

by Alexander S. Kechris , Alain Louveau

• ISBN13: 9780521358118
• ISBN10: 0521358116
• Format: Paperback
• Copyright: 1987-11-27
• Publisher: Cambridge University Press
• More Book Details

Summary

The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis.

Table of Contents

Introduction	1	(16)
About This Book	17	(4)
Trigonometric Series and Sets of Uniqueness	21	(30)
Trigonometric and Fourier Series	21	(2)
The problem of uniqueness	23	(2)
The Riemann theory and the Cantor Uniqueness Theorem	25	(8)
The Rajchman multiplication theory. Examples of perfect sets of uniqueness	33	(8)
Countable unions of closed sets of uniqueness	41	(5)
Four classical problems	46	(5)
The Algebra a of Functions with Absolutely Convergent Fourier Series, Pseudofunctions and Pseudomeasures	51	(29)
The spaces PF, A and PM	51	(5)
Some basic facts about A	56	(6)
Supports of pseudomeasures	62	(7)
Description of closed U-sets in terms of pseudofunctions	69	(6)
Rajchman measures and extended uniqueness sets	75	(5)
Symmetric Perfect Sets and the Salem-Zygmund Theorem	80	(24)
H(n)-sets	80	(4)
Pisot numbers	84	(3)
Symmetric and homogeneous perfect sets	87	(3)
The Salem-Zygmund Theorem	90	(14)
Classification of the Complexity of U	104	(35)
some descriptive set theory	104	(13)
The theorem of Solovay and Kaufman	117	(14)
On σ-ideals of closed sets in compact, metrizable spaces	131	(8)
The Piatetski-Shapiro Hierarchy of U-Sets	139	(54)
Γ1 1-ranks on Γ1 1 sets	140	(10)
Ranks for subspaces of Banach spaces	150	(11)
The tree-rank and the R-rank	161	(13)
The Piatetski-Shapiro rank on U	174	(8)
The class U' of uniqueness sets of rank 1	182	(11)
Decomposing U-Sets into Simpler Sets	193	(38)
Borel bases for σ-ideals of closed sets	193	(18)
The class U1 and the decomposition theorem of Piatetski-Shapiro	211	(9)
The Borel Basis Problem for U and relations between U, U1 and U0	220	(11)
The Shrinking Method, The Theorem of Korner and Kaufman, and the Solution to the Borel Basis Problem for U	231	(35)
Sets of interior uniqueness	231	(8)
Approximating M-sets by HO-sets	239	(4)
Helson sets of multiplicity	243	(17)
The solution to the Borel Basis Problem	260	(6)
Extended Uniqueness Sets	266	(43)
The class U'O	266	(8)
The existence of a Borel basis for UO and its associated rank	274	(15)
The solution to the Category Problem, and other applications	289	(11)
The class U1 revisited	300	(9)
Characterizing Rajchman Measures	309	(19)
A theorem of Mokobodzki in measure theory	309	(13)
W-sets and Lyons' characterization of Rajchman measures	322	(6)
Sets of Resolution and Synthesis	328	(21)
Sets of resolution	328	(13)
Sets of synthesis	341	(8)
List of Problems	349	(4)
References	353	(6)
Symbols and Abbreviations	359	(4)
Index	363