On the structure of finite level and \omega-decomposable Borel functions

TL;DR

This paper thoroughly characterizes finite level Borel functions, explores their decomposability into continuous functions, and applies these insights to Banach space theory, advancing the understanding of Borel function structure.

Contribution

It provides a complete description of finite level Borel classes, elementary proofs of key facts, and new results on extomega-decomposability, extending classical theorems to all finite levels.

Findings

Not all Borel functions are countably extSigma^0_eta-measurable.

Finite level Borel functions can be decomposed into simpler functions.

Applications to Banach space theory demonstrate practical relevance.

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 \Sigma^0_\alpha-measurable functions (for every fixed 1 \leq \alpha < \omega_1). Moreover, we present some results concerning those Borel functions which are \omega-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.