Functions of the first Baire class with values in Banach spaces
HTML articles powered by AMS MathViewer
- by Charles Stegall PDF
- Proc. Amer. Math. Soc. 111 (1991), 981-991 Request permission
Abstract:
We characterize functions of the first Baire class with values in Banach spaces and give a short self-contained proof of a result more general than the following: If $T$ is a complete metric space, $X$ is a Banach space, and $\Phi :T \to \wp (X)$ (the power set of $X$) is a mapping that is use in the weak topology then $\Phi$ has a selector of the first Baire class. This extends some results of Hansell, Jayne, Rogers, and Talagrand.References
- J. Bourgain, Compact sets of first Baire class, Bull. Soc. Math. Belg. 29 (1977), no. 2, 135–143. MR 470931
- J. Bourgain, D. H. Fremlin, and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), no. 4, 845–886. MR 509077, DOI 10.2307/2373913
- Jens Peter Reus Christensen, Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), no. 3, 455–461. MR 612739, DOI 10.1090/S0002-9939-1981-0612739-1
- Zdeněk Frolík, A measurable map with analytic domain and metrizable range is quotient, Bull. Amer. Math. Soc. 76 (1970), 1112–1117. MR 265539, DOI 10.1090/S0002-9904-1970-12584-8 —, Baire spaces and some generalizations of complete metric spaces, Czech. Math. J. 20 (1970), 406-467.
- R. W. Hansell, First class selectors for upper semicontinuous multifunctions, J. Funct. Anal. 75 (1987), no. 2, 382–395. MR 916758, DOI 10.1016/0022-1236(87)90102-9
- R. W. Hansell, J. E. Jayne, and M. Talagrand, First class selectors for weakly upper semicontinuous multivalued maps in Banach spaces, J. Reine Angew. Math. 361 (1985), 201–220. MR 807260
- F. Hausdorff, Mengenlehre, Dover Publications, New York, N.Y., 1944 (German). MR 0015445
- J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), no. 1-2, 41–79. MR 793237, DOI 10.1007/BF02392537 K. Kuratowski, Topologie, PWN, Warszawa, 1958.
- Henri Lebesgue, Oeuvres scientifiques (en cinq volumes). Vol. III, Université de Genève, Institut de Mathématiques, Genève, 1972 (French). Sous la rédaction de François Châtelet et Gustave Choquet. MR 0389523
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515–531. MR 370466 C. Stegall, Applications of descriptive topology in functional analysis, Universität Linz, 1985; 2nd ed., 1987. —, Notes, 1976, unpublished.
- Charles Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodým property, Israel J. Math. 29 (1978), no. 4, 408–412. MR 493268, DOI 10.1007/BF02761178
- Charles Stegall, Generalizations of a theorem of Namioka, Proc. Amer. Math. Soc. 102 (1988), no. 3, 559–564. MR 928980, DOI 10.1090/S0002-9939-1988-0928980-0
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 981-991
- MSC: Primary 26A21; Secondary 47H04
- DOI: https://doi.org/10.1090/S0002-9939-1991-1019283-7
- MathSciNet review: 1019283