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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A classification of Baire class $ 1$ functions


Authors: A. S. Kechris and A. Louveau
Journal: Trans. Amer. Math. Soc. 318 (1990), 209-236
MSC: Primary 26A21; Secondary 04A15, 26A24, 54C50
MathSciNet review: 946424
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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 $ {\mathcal{B}_1}$ in a transfinite hierarchy $ \mathcal{B}_1^\xi (\xi < {\omega _1})$ of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of $ \Delta _2^0$ 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 $ f$ are introduced and used in this study.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0946424-3
Article copyright: © Copyright 1990 American Mathematical Society