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How can we recover Baire class one functions?

Published online by Cambridge University Press:  26 February 2010

Dominique Lecomte
Affiliation:
Equipe d'analyse de l'Université Paris 6, Tour 46-0, boîte 186, 4, place Jussieu, 75 252 Paris cedex 05, France. E-mail: [email protected]
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Abstract

Let X and Y be separable metrizable spaces, and f: X→Y a function. It is wished to recover f from its values on a small set via a simple algorithm. It is shown that this is possible if f is Baire class one, and in fact a characterization is obtained. This leads to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

Type
Research Article
Copyright
Copyright © University College London 2003

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References

[Bol]Bourbaki, N., Topohgiegenerate, ch. 1-4, Hermann, Paris (1974).Google Scholar
[Bo2]Bourbaki, N., Topologie generate, ch. 5-10, Hermann, Paris (1974).Google Scholar
[Bl]Bourgain, J., Some remarks on compact sets of first Baire class. Bull. Soc. Math. Belg.. 30(1978), 310.Google Scholar
[B2]Bourgain, J., On convergent sequences of continuous functions. Bull. Soc. Math. Beig. Sér. B, 32 (1980), 235249.Google Scholar
[DE]Darji, U. B. and Evans, M. J., Recovering Baire one functions. Mathematika. 42 (1995). 4348.Google Scholar
[DS]Dunford, N. and Schwartz, J. T., Linear operators, Part 1, Interscience Publishers. Inc.. New-York (1964).Google Scholar
[FV]Freiling, C. and Vallin, R. W., Simultaneous recovery of Baire one functions. Real Analysis Exchange 22, Vol. 1 (1996/1997), 346349.CrossRefGoogle Scholar
[Ke]Kechris, A. S., Classical Descriptive Set Theory, Springer Verlag, New-York (1995).Google Scholar
[KL]Kechris, A. S. and Louveau, A., A classification of Baire class one functions, Trans. Amer. Math. Soc, 318, 1 (1990), 209236.CrossRefGoogle Scholar
[Ku]Kuratowski, K., Topology, Vol. 1, Academic Press, New–York London (1996).Google Scholar
[LTZ]Lee, P. Y., Tang, W. K. and Zhao, D., An equivalent definition of the functions of the first Baire class, Proc. Amer. Math. Soc, 129, 8 (2000). 22732275.Google Scholar
[LI]Louveau, A., A separation theorem for sets. Trans. Amer. Math. Soc. 260 (1980). 363378.Google Scholar
[L2]Louveau, A., Some results in the Wadge hierarchy of Borel sets. Cabal Sem. 79-81 (Kechris, A. S, Mauldin, D. A., Moschorakis, Y. N., eds) Lecture Notes in Math. 1019 Sprinaer–Verlag (1983), 2855.Google Scholar
[LSR]Louveau, A. and Raymond, J. Saint, The strength of Borel Wadge determinacy. Cabal Sem. 81-85, Lecture Notes in Math. 1333 Springer–Verlag (1988). 130.Google Scholar
[M]Moschovakis, Y. N., Descriptive Set Theory, North–Holland (1980).Google Scholar
[OR]Odell, E. and Rosenthal, H. P., A double-dual characterization of separable Banach spaces containing l1 Israel J. Math., 20 (1975). 375384.Google Scholar
[T]TodorCeviĆ, S., Compact subsets of the first Baire class, J. Amer. Math. Soc (4)12(1999). 11791212.Google Scholar