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On the Structure of Finite Level and ω-Decomposable Borel Functions

Published online by Cambridge University Press:  12 March 2014

Luca Motto Ros
Luca Motto Ros*
Affiliation:
Albert–Ludwigs–Universität Freiburg, Mathematisches Institut – Abteilung für Mathematische Logik, Eckerstraße 1, D-79104 Freiburg im Breisgau, Germany, E-mail: luca.motto.ros@math.uni-freiburg.de
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Abstract

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

Keywords

03E1554H0526A2154C1054E4046J1046B04Finite level Borel functioncountably continuous functionBaire class of functionsdecomposable function
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1257 - 1287
Copyright
Copyright © Association for Symbolic Logic 2013

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