Baire functions, borel sets, and ordinary function systems

Abstract

Ifis a family of real-valued functions defined on a set X, then there is a smallest family, B(qt), of real-valued functions defined on X which contains ~ and which is closed under the process of taking pointwise limits of sequences from B(~). This family is called the Baire system generated by cb. One method of generating B(~) from q5 is by iteration of the operation of pointwise limits of sequences: Let ~0 be ~ and for each ordinal c~ > 0, let q5 be the family of all pointwise limits of sequences from (J~<~ q)~. Then q5~1 -~ ~1+1 = B(qt), where ~o 1 is the first un- countable ordinal. This system was described by Ren6 Baire in this th6se, published in 1899 (2). This paper is meant to be an exposition of some of the main results concerning this process that have been obtained since then. The second section concerns itself with some properties of the classes ~, under the assumption that the family ~ forms a lattice. In the third section, a development of the relationship between the classes q~ and Borel type sets which are inverse image sets of functions in q~ is given. That section concludes with Hausdorff's notion of an ordinary function system. These systems are completely characterized by their inverse image sets and yield appealing coextensive processes of generating the Baire system and of generating a certain Borel system (a-algebra) of sets. The Baire order of a family of functions # is the first ordinal ~ such that ~ ~ ~+1 • The Baire order problem for C(X), the space of real- valued continuous functions on a topological space X, is studied in the fourth section. Two proofs, due to Lebesgue, are given to show that the Baire order of C(0, 1) is oJ 1 . Later in this section it is shown that the Baire order of C(X), with X compact and Te, distinguishes those spaces which contain perfect sets from those which do not (dispersed spaces); the Baire order of the dispersed spaces being 0 or I and the others ~o 1 . In the last section the Baire order of various families of functions 418