A Classification of Baire Class 1 Functions

Abstract

We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class 1 functions B_1 in a transfinite hierarchy B^ξ_1 ξ < ω_1) of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of Δ^0_2 sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function ƒ are introduced and used in this study.

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