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