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

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.

Decomposing Functions of Baire Class on Polish Spaces

“…The Decomposability Conjecture (cf. [1,8,7]). Let X , Y be separable metrizable spaces with X analytic, and let f : X → Y .…”

“…In recent years, there have been a number of remarkable progresses on the Decomposability Conjecture, cf. [2,6,8,7,9]. Most recently, Gregoriades-Kihara-Ng [2] proved…”

“…It is trivial to see that, for ≥ , f ∈ dec(Σ 0 , ∆ 0 ) implies the Σ 0measurability of f. It is also well known that, for any Σ 0 -measurable function f with > , we have f ∈ dec(Σ 0 ) ⇐⇒ f ∈ dec(Σ 0 , ∆ 0 +1 ) (cf. [8,Proposition 4.5]). The following proposition give some well known properties which will be used again and again in the rest of this article.…”

Decomposing Borel functions using the Shore-Slaman join theorem

“…A remarkable theorem proved by Jayne-Rogers [15] states that the Σ 2,2 functions are precisely the ∆ 0 2 -piecewise continuous functions, where for a class Γ of Borel sets and a class F of Borel functions, we say that a function is Γ-piecewise F (denoted by the symbol dec α F if Γ is a delta class ∆ 0 α ) if it is decomposable into countably many F -functions with Γ domains (see also [17] for an alternative proof). Subsequently, Solecki [31] proved a dichotomy (see also [21,24,26]) sharpening the Jayne-Rogers theorem by using the Gandy-Harrington topology from effective descriptive set theory.…”

“…More recently, a significant breakthrough was made by Semmes [28], who used Wadge-like infinite two-player games and priority arguments to show that on the zero-dimensional Polish space ω ω , the Σ 3,3 functions are precisely the ∆ 0 3 -piecewise continuous functions, and the Σ 2,3 functions are precisely the ∆ 0 3 -piecewise Σ 0 2measurable (i.e., Σ 1,2 ) functions. Countable decomposability at all finite levels of Borel hierarchy has been studied by Pawlikowski-Sabok [24] and Motto Ros [21]. Naturally, many researchers expected that the Jayne-Rogers theorem and the Semmes theorem could be generalized to all finite levels of the hierarchy of Borel functions (see Andretta [1], Semmes [28] Decomposability Conjecture.…”