A Classification of Baire Class 1 Functions

Abstract

We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence.This leads to a natural classification of the class of bounded Baire class 1 functions Ml in a transfinite hierarchy q7 (4 < wol) of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of AO sets via transfinite differences of closed sets can be carried out.The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function f are introduced and used in this study.Let E be a compact metrizable space.Our aim is to analyze, from a descriptive theoretic point of view, the set of Baire class 1 functions on E. Various ordinal assignments have been proposed for Baire class 1 functions, especially in Banach space theory.In the first part of this work, we will associate to each Baire class 1 function on E three different ordinal ranks, each corresponding to one of the siandard equivalent descriptions of Baire class 1 functions, in terms of (i) complexity of inverse images of open sets, (ii) existence of continuity points on closed sets, and (iii) as limits of sequences of continuous functions.We will then show the essential equivalence of these ranks for bounded Baire class 1 functions, thus establishing a "quantitative" version of the equivalence of the above descriptions.This yields, in ?2, a natural classification of the set I(E) of bounded Baire class 1 functions into a transfinite hierarchy of Banach algebras qj(E), 4 < O I, that we call (bounded) functions of small Baire class 4 .In ?3, we study the first level wI of this hierarchy, showing that it corresponds to a previously considered notion, the so-called strict Baire class 1 functions, the uniform limits of differences of bounded semicontinuous functions.We also define a natural rank on the subset DBSC of differences of bounded semicontinuous functions, and a rank on functions not in DBSC, and show that these two ranks are unbounded below w1, on DBSC and on l \ DBSC respectively.In ?4,we give two other different approaches to our classification.The first one uses the notion of (transfinite) alternating sums of usc functions, and allows