Filter descriptive classes of Borel functions

Abstract

Abstract. We first prove that given any analytic filter F on ω the set of all functions f on 2 ω which can be represented as the pointwise limit relative to F of some sequence (fn)n∈ω of continuous functions (f = limF fn), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of F. We discuss several structural properties of this rank. For example, we prove that any free Π 0 4 filter is of rank 1.

Ranks of F-limits of filter sequences

“…This is apparent from the following theorem of Debs and Saint Raymond [2]: Theorem 1.1. Let F be an analytic filter and α < ω 1 be a countable ordinal.…”

“…According to Proposition 4.4 from [2], rk(G) ≥ α + 1. Notice that the F -Fubini sum of the family (F i ) i∈I can be represented as an F -limit of the family of filters…”

“…The first property in the above proposition is a simple consequence of the fact that the canonical involution A  → A c is a homeomorphism. Proofs of the remaining properties can be found in [2].…”

“…We omit the definition of filters N γ for ω ≤ γ < ω 1 ; however, it can be found in [2]. Theorem 2.2 can also be generalized: for every α < ω 1 , the filter N α has rank α (cf.…”

“…[1]), every analytic filter has countable rank. The concept of filter rank was introduced by Debs and Saint Raymond [2], although it was also studied by Solecki [3] in the context of a question raised by Dobrowolski and Marciszewski [4]. Clearly, for every filter F , its rank rk(F ) is unique.…”