Ranks on the Baire class 𝜉 functions

Elekes, Márton; Kiss, Viktor; Vidnyánszky, Zoltán

doi:10.1090/tran/6764

Ranks on the Baire class $\xi$ functions
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by Márton Elekes, Viktor Kiss and Zoltán Vidnyánszky PDF

Trans. Amer. Math. Soc. 368 (2016), 8111-8143 Request permission

Abstract:

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $\xi$ functions and generalize most of the results from the Baire class $1$ case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so-called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $\omega _1$. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions.

References

Spiros A. Argyros, Gilles Godefroy, and Haskell P. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1007–1069. MR 1999190, DOI 10.1016/S1874-5849(03)80030-X
J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belg. Sér. B 32 (1980), no. 2, 235–249. MR 682645
Márton Elekes and Miklós Laczkovich, A cardinal number connected to the solvability of systems of difference equations in a given function class, J. Anal. Math. 101 (2007), 199–218. MR 2346545, DOI 10.1007/s11854-007-0008-y
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 10.1090/S0002-9947-1930-1501551-9
Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated from the German by John R. Aumann et al. MR 0141601
R. Haydon, E. Odell, and H. Rosenthal, On certain classes of Baire-$1$ functions with applications to Banach space theory, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 1–35. MR 1126734, DOI 10.1007/BFb0090209
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
A. S. Kechris and A. Louveau, A classification of Baire class $1$ functions, Trans. Amer. Math. Soc. 318 (1990), no. 1, 209–236. MR 946424, DOI 10.1090/S0002-9947-1990-0946424-3
Miklós Laczkovich, Decomposition using measurable functions, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 6, 583–586 (English, with English and French summaries). MR 1411046
Miklós Laczkovich, Operators commuting with translations, and systems of difference equations, Colloq. Math. 80 (1999), no. 1, 1–22. MR 1684566, DOI 10.4064/cm-80-1-1-22
Z. Zalcwasser, Sur une propriété du champs des fonctions continues, Studia Math. 2 (1930), 63–67.