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Though the search for good selectors dates back to the early twentieth century, selectors play an increasingly important role in current research. This book is the first to assemble the scattered literature into a coherent and elegant presentation of what is known and proven about selectors--and what remains to be found.
The authors focus on selection theorems that are related to the axiom of choice, particularly selectors of small Borel or Baire classes. After examining some of the relevant work of Michael and Kuratowski & Ryll-Nardzewski and presenting background material, the text constructs selectors obtained as limits of functions that are constant on the sets of certain partitions of metric spaces. These include selection theorems for maximal monotone maps, for the subdifferential of a continuous convex function, and for some geometrically defined maps, namely attainment and nearest-point maps.
Assuming only a basic background in analysis and topology, this book is ideal for graduate students and researchers who wish to expand their general knowledge of selectors, as well as for those who seek the latest results.
List of contents
Preface vii
Introduction ix
Chapter 1. Classical results 1
1.1 Michael's Continuous Selection Theorem 1
1.2 Results of Kuratowski and Ryll-Nardzewski 8
1.3 Remarks 13
Chapter 2. Functions that are constant on the sets of a disjoint discretely o-decomposable family of F
s-sets 19
2.1 Discretely o-Decomposable Partitions of a Metric Space 19
2.2 Functions of the First Borel and Baire Classes 25
2.3 When is a Function of the First Borel Class also of the First Baire Class? 39
2.4 Remarks 42
Chapter 3. Selectors for upper semi-continuous functions with non-empty compact values 43
3.1 A General Theorem 45
3.2 Special Theorems 53
3.3 Minimal Upper Semi-continuous Set-valued Maps 53
3.4 Remarks 57
Chapter 4. Selectors for compact sets 65
4.1 A Special Theorem 67
4.2 A General Theorem 69
4.3 Remarks 88
Chapter 5. Applications 91
5.1 Monotone Maps and Maximal Monotone Maps 95
5.2 Subdifferential Maps 101
5.3 Attainment Maps from X* to X 106
5.4 Attainment Maps from X to X* 107
5.5 Metric Projections or Nearest Point Maps 108
5.6 Some Selections into Families of Convex Sets 110
5.7 Example 118
5.8 Remarks 122
Chapter 6. Selectors for upper semi-continuous set-valued maps with nonempty values that are otherwise arbitrary 123
6.1 Diagonal Lemmas 124
6.2 Selection Theorems 127
6.3 A Selection Theorem for Lower Semi-continuous Set-valued Maps 138
6.4 Example 140
6.5 Remarks 144
Chapter 7. Further applications 147
7.1 Boundary Lemmas 149
7.2 Duals of Asplund Spaces 151
7.3 A Partial Converse to Theorem 5.4 156
7.4 Remarks 159
Bibliography 161
Index 165
About the author
John E. Jayne & C. Ambrose Rogers
Summary
Focuses on selection theorems that are related to the axiom of choice, particularly selectors of small Borel or Baire classes. This text constructs selectors obtained as limits of functions that are constant on the sets of certain partitions of metric spaces. It is aimed at students and researchers who wish to expand their knowledge of selectors.
