Selectors

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 Fs-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