Keele Research Repository

Let S and X be any two sets; then a mapping T which assigns to each point t in S a set T(t) of points in X is called a multifunction (or set-valued function) from S into X. A selector for T is a function f from S into X such that f(t) belongs to the set r(t) for each t. This thesis contains a systematic study of multifunctions, especially measurable multifunctions, and a number of instances are given where a multifunction T has a selector which inherits the good properties of T, or at least is not much worse. The problem of proving that selectors exist can be approached from more than one direction; in particular the class of multifunctions of Souslin type is introduced. This class is comprehensive, containing the kinds of measurable multifunction most commonly studied previously, it is closed under the usual operations of analysis and set-theory, and yet it is well-supplied with measurable selectors.