A classification of Baire class 1 functions

Kechris, A. S.; Louveau, A.

doi:10.1090/S0002-9947-1990-0946424-3

A classification of Baire class $1$ functions
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by A. S. Kechris and A. Louveau PDF

Trans. Amer. Math. Soc. 318 (1990), 209-236 Request permission

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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Sur une propriété du champ des fonctions continues

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