Filter descriptive classes of Borel functions

Gabriel Debs¹ ; Jean Saint Raymond²

¹ Analyse Fonctionnelle Institut de Mathématique de Jussieu Boîte 186 4, place Jussieu 75252 Paris Cedex 05, France
² Analyse Fonctionnelle Institut de Mathématique de Jussieu Boîte 186 4 place Jussieu F- 75252 Paris Cedex 05, France

Fundamenta Mathematicae, Tome 204 (2009) no. 3, pp. 189-213

Résumé

\We first prove that given any analytic filter ${\cal F}$ on $\omega$ the set of all functions $f$ on ${\bf 2}^\omega$ which can be represented as the pointwise limit relative to ${\cal F}$ of some sequence $ (f_{n})_{n\in\omega}$ of continuous functions ($f=\lim_{\cal F} f_n$), is exactly the set of all Borel functions of class $\xi$ for some countable ordinal $\xi$ that we call the rank of ${\cal F}$. We discuss several structural properties of this rank. For example, we prove that any free $\Pi^0_ 4$ filter is of rank 1.

Détail

DOI : 10.4064/fm204-3-1

Keywords: first prove given analytic filter cal omega set functions omega which represented pointwise limit relative cal sequence omega continuous functions lim cal exactly set borel functions class countable ordinal call rank cal discuss several structural properties rank example prove filter rank