Classification of bounded Baire class $\xi $ functions
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 semicontinuous 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 necessarily 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. (2014).