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

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.

J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belg. Sér. B 32 (1980), no. 2, 235–249. MR 682645

G. Choquet, Remarques à propos de la démonstration d’unicité de P.-A. Meyer, (appendice), Séminaire de Théorie du Potentiel, $6^{e}$ année, exposé no. 8, Inst. Hautes Etudes Sci. Paris, 1961-62.

D. C. Gillespie and W. A. Hurwitz, On sequences of continuous functions having continuous limits, Trans. Amer. Math. Soc. 32 (1930), no. 3, 527–543. MR 1501551, DOI https://doi.org/10.1090/S0002-9947-1930-1501551-9

R. Haydon, E. Odell and H. P. Rosenthal, Certain subclasses of Baire-$1$ functions with Banach space applications, circulated notes.

Alexander S. Kechris and W. Hugh Woodin, Ranks of differentiable functions, Mathematika 33 (1986), no. 2, 252–278 (1987). MR 882498, DOI https://doi.org/10.1112/S0025579300011244
E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing $l^{1}$, Israel J. Math. 20 (1975), no. 3-4, 375–384. MR 377482, DOI https://doi.org/10.1007/BF02760341
G. Petruska and M. Laczkovich, Baire $1$ functions, approximately continuous functions and derivatives, Acta Math. Acad. Sci. Hungar. 25 (1974), 189–212. MR 379766, DOI https://doi.org/10.1007/BF01901760

Z. Zalcwasser, Sur une propriété du champ des fonctions continues, Studia Math. 2 (1930), 63-67.