A Course on Borel Sets (Google eBook)

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Springer Science & Business Media, Apr 13, 1998 - Mathematics - 261 pages
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A Course on Borel sets provides a thorough introduction to Borel sets and measurable selections and acts as a stepping stone to descriptive set theory by presenting important techniques such as universal sets, prewellordering, scales, etc. It is well suited for graduate students exploring areas of mathematics for their research and for mathematicians requiring Borel sets and measurable selections in their work. It contains significant applications to other branches of mathematics and can serve as a self- contained reference accessible by mathematicians in many different disciplines. It is written in an easily understandable style and employs only naive set theory, general topology, analysis, and algebra. A large number of interesting exercises are given throughout the text.
  

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Contents

Cardinal and Ordinal Numbers
1
12 Order of Infinity
4
13 The Axiom of Choice
7
14 More on Equinumerosity
11
15 Arithmetic of Cardinal Numbers
13
16 WellOrdered Sets
15
17 Transfinite Induction
18
18 Ordinal Numbers
21
44 The First Separation Theorem
147
45 OnetoOne Borel Functions
150
46 The Generalized First Separation Theorem
155
47 Borel Sets with Compact Sections
157
48 Polish Groups
160
49 Reduction Theorems
164
410 Choquet Capacitability Theorem
172
411 The Second Separation Theorem
175

19 Alephs
24
110 Trees
26
111 Induction on Trees
29
112 The Souslin Operation
31
113 Idempotence of the Souslin Operation
34
Topological Preliminaries
39
22 Polish Spaces
52
23 Compact Metric Spaces
57
24 More Examples
63
25 The Baire Category Theorem
69
26 Transfer Theorems
74
Standard Borel Spaces
80
32 BorelGenerated Topologies
91
33 The Borel Isomorphism Theorem
94
34 Measures
100
35 Category
107
36 Borel Pointclasses
115
Analytic and Coanalytic Sets
127
42 𝚺11 and 𝚷11 Complete Sets
135
43 Regularity Properties
141
412 CountabletoOne Borel Functions
178
Selection and Uniformization Theorems
183
51 Preliminaries
184
52 Kuratowski and RyllNardzewskis Theorem
189
53 Dubins Savage Selection Theorems
194
54 Partitions into Closed Sets
195
55 Von Neumanns Theorem
198
56 A Selection Theorem for Group Actions
200
57 Borel Sets with Small Sections
204
58 Borel Sets with Large Sections
206
59 Partitions into G𝜎 Sets
212
510 Reflection Phenomenon
216
511 Complementation in Borel Structures
218
512 Borel Sets with σCompact Sections
219
513 Topological Vaught Conjecture
227
514 Uniformizing Coanalytic Sets
236
References
241
Glossary
250
Index
253
Copyright

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Page iv - Mathematician. 2nd ed. 6 HUGHES/PIPER. Projective Planes. 7 SERRE. A Course in Arithmetic. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory.
Page 244 - Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge Univ.

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About the author (1998)

Srivastava-Indian Statistical Institute, Calcutta, India

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