Selection and Uniformization Theorems

In this chapter we present some measurable selection theorems. Selection theorems are needed in several branches of mathematics such as probability theory, stochastic processes, ergodic theory, mathematical statistics ([17], [34], [89], [18], etc.), functional analysis, harmonic analysis, representation theory of groups and C*-algebras ([4], [6], [7], [35], [36], [37], [40], [50], [54], [72], [73], [124], etc.), game theory, gambling, dynamic programming, control theory, mathematical economics ([55], [78], etc.). Care has been taken to present the results in such a way that they are readily applicable in a variety of situtations. It is impossible to present a satisfactory account of applications in a book of this size. We shall be content with giving some applications that do not require much background beyond what has been developed in this book. From time to time we give some references, where interested readers will find more applications.