Classification of bounded Baire class $\xi $ functions

Viktor Kiss¹

¹ Department of Analysis Eötvös Loránd University Pázmány Péter Sétány 1/c H-1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 236 (2017) no. 2, pp. 141-160.

Résumé

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).

Détail

DOI : 10.4064/fm194-1-2016

Keywords: kechris louveau showed each real valued bounded baire class nbsp function defined compact metric space written alternating sum decreasing countable transfinite sequence upper semicontinuous functions moreover length shortest sequence essentially the value certain natural ranks defined baire class nbsp functions introduced notion pseudouniform convergence generate classes bounded baire class nbsp functions others main paper generalize their results baire class functions proofs through essential first obtain similar results baire class nbsp functions defined necessarily compact polish spaces using these classifications bounded baire class functions define natural ranks these classes these ranks essentially coincide those defined elekes nbsp nbsp