Functions of the first Baire class with values in Banach spaces

Stegall, Charles

doi:10.1090/S0002-9939-1991-1019283-7

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

20

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

Topologie

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

Applications of descriptive topology in functional analysis

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