Selectors

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. Contents 6 Preface 8 Introduction 10 Chapter 1. Classical results 16 1.1 Michael’s Continuous Selection Theorem 16 1.2 Results of Kuratowski and Ryll-Nardzewski 23 1.3 Remarks 28 Chapter 2. Functions that are constant on the sets of a disjoint discretely σ-decomposable family of F[sub(σ)]-sets 34 2.1 Discretely σ-Decomposable Partitions of a Metric Space 34 2.2 Functions of the First Borel and Baire Classes 40 2.3 When is a Function of the First Borel Class also of the First Baire Class? 54 2.4 Remarks 57 Chapter 3. Selectors for upper semi-continuous functions with non-empty compact values 58 3.1 A General Theorem 60 3.2 Special Theorems 68 3.3 Minimal Upper Semi-continuous Set-valued Maps 68 3.4 Remarks 72 Chapter 4. Selectors for compact sets 80 4.1 A Special Theorem 82 4.2 A General Theorem 84 4.3 Remarks 103 Chapter 5. Applications 106 5.1 Monotone Maps and Maximal Monotone Maps 110 5.2 Subdifferential Maps 116 5.3 Attainment Maps from X[sup(*)] to X 121 5.4 Attainment Maps from X to X[sup(*)] 122 5.5 Metric Projections or Nearest Point Maps 123 5.6 Some Selections into Families of Convex Sets 125 5.7 Example 133 5.8 Remarks 137 Chapter 6. Selectors for upper semi-continuous set-valued maps with nonempty values that are otherwise arbitrary 138 6.1 Diagonal Lemmas 139 6.2 Selection Theorems 142 6.3 A Selection Theorem for Lower Semi-continuous Set-valued Maps 153 6.4 Example 155 6.5 Remarks 159 Chapter 7. Further applications 162 7.1 Boundary Lemmas 164 7.2 Duals of Asplund Spaces 166 7.3 A Partial Converse to Theorem 5.4 171 7.4 Remarks 174 Bibliography 176 Index 180 A 180 B 180 C 180 D 180 E 180 F 180 G 181 H 181 J 181 K 181 L 181 M 181 N 181 O 181 P 181 R 181 S 182 T 182 V 182 W 182 Z 182 Many selection theorems have been proved; many of the earlier ones are called uniformization theorems.