A Classification of Baire Class 1 Functions

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 B1in a transfinite hierarchy$\mathscr{B}^\zeta_1 (\zeta < \omega_1)$of "small"...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1990, Vol.318 (1), p.209-236
Main Authors: Kechris, A. S., Louveau, A.
Format: Article
Language:English
Subjects:
Alternating series
Approximation
Banach space
Continuous functions
Decreasing sequences
Equivalence relation
Mathematical functions
Mathematics
Perceptron convergence procedure
Series convergence
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Summary: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 B1in a transfinite hierarchy$\mathscr{B}^\zeta_1 (\zeta < \omega_1)$of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of Δ0 2sets 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/s0002-9947-1990-0946424-3