1981/07/01 by John P. Burgess
Mathematics · Business, Management and Accounting · Decision Sciences · #Fuzzy Systems and Optimization #Organizational Management and Leadership #Decision-Making and Behavioral Economics
paper · pdf · doi:10.1305/ndjfl/1093883456
Selector theory as surveyed in [13] and [14] deals with the following problem (instances of which arise in control theory, probability, mathematical economics, operator theory, etc.):We are given a multifunction F between reasonable spaces T and X (a map assigning each t e T a nonempty Ft) C X) and seek an ordinary function / from T to X with acceptable measurability properties constituting a selector for F (satisfying f(t) e F(t) for all t).Of course, the Axiom of Choice says that a selector exists; but to get a measurable one, we need to impose hypotheses on F and choose "carefully".The past few years have seen much progress (cf.[ 10], [11], [ 14]) on the Borel case of the selector problem.In this case we assume X is a Polish topological space (one admitting a countable basis and a complete metric) and T at least a Suslin space (homeomorph of an analytic subspace of a Polish space).Our goal is to find weak hypotheses on F guaranteeing the existence of a Borel-measurable selector /(one for which fThe present paper* shows that substantial improvements of existing results on the Borel selector problem can be achieved through application of ideas developed by Vaught in his prize-winning studies [12] on the model theory of infinitary logic.The precise statement of the result obtained is given in Section 3 below.Thanks to certain counterexamples, we can say that this result is in many ways "best possible".Selector theory is thus a relatively down-to-earth area of mathematics where methods from modern logical research can be fruitfully applied.