Some selection theorems and problems

Abstract

Let I be the closed unit interval, [0,i]. Let B be a Borel subset of I×I such that for each X,Bx=~Y:(x,y) EB}~ ~. Using the axiom of choice, we find that the Borel set B contains a uniformization (= the graph of some function f mapping I onto I). The question was raised concerning how nice or describable the function f is in the famous letters exchanged among Baire, Borel, Hadamard and Lebesgue [I]. Novikov gave the first example of a Borel subset of I×I which does not possess a Borel uniformization [2]. Kondo proved that every such Borel set B possesses a uniformization which is coanalytic [3]. Yankov [4] and von Neumann [5] proved that B contains the graph of a function f which is measurable with respect to the ~-algebra generated by the analytic subsets of I. In fact, they proved this result assuming only that B is an analytic set. Whether every Borel set B possesses a uniformization which is the difference of two coanalytic sets seems to be an unsolved problem.