Classification of bounded Baire class $\xi$ functions

TL;DR

This paper generalizes the classification of bounded Baire class 1 functions to Baire class , establishing new ranks and decompositions that extend previous results from compact spaces to more general Polish spaces.

Contribution

It introduces a new classification framework for Baire class  functions, extending existing results to non-compact Polish spaces and defining ranks that align with prior work.

Findings

Established a decomposition of Baire class  functions into transfinite sequences

Defined natural ranks on these classes that match existing ranks by Elekes et al.

Extended classification results from compact to general Polish spaces.

Abstract

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions. Moreover, the length of the shortest such sequence is essentially the same as the value of certain natural ranks they defined on the Baire class 1 functions. They also introduced the notion of pseudouniform convergence to generate some classes of bounded Baire class 1 functions from others. The main aim of this paper is to generalize their results to Baire class $\xi$ functions. For our proofs to go through, it was essential to first obtain similar results for Baire class 1 functions defined on not necessary compact Polish spaces. Using these new classifications of bounded Baire class $\xi$ functions, one can define natural ranks on these classes. We show that these ranks essentially coincide with those defined by Elekes et. al.