A classification of Baire class 1 functions. (English) Zbl 0692.03031

Kechris, A. S.; Louveau, A.

Trans. Am. Math. Soc. 318, No. 1, 209-236 (1990).

The authors study three variants of ordinal ranks for functions of Baire class 1 and show that for bounded functions these are essentially equivalent. Thus one obtains a hierarchy of Banach algebras \(B_ 1^{\xi}\) \((\xi <\omega_ 1)\) of bounded Baire-class 1 functions with the first level \(B^ 1_ 1\) corresponding to the strict Baire class 1 functions. Two other approaches to the authors’ classification are given: the first is similar to the Hausdorff-Kuratowski analysis of \(\Delta^ 0_ 2\) sets via transfinite differences of closed sets and uses alternating sums of usc functions, the second uses a notion of pseudo- uniform convergence. Finally the problem of optimal convergence for derivatives is studied.

Reviewer: K.Gloede

MSC:

03E15	Descriptive set theory
46B99	Normed linear spaces and Banach spaces; Banach lattices

Keywords:

ordinal ranks for functions of Baire class 1; bounded functions; hierarchy of Banach algebras; Hausdorff-Kuratowski analysis of \(\Delta ^ 0_ 2\) sets; transfinite differences of closed sets; alternating sums of usc functions; pseudo-uniform convergence; optimal convergence for derivatives