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.
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.
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
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
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