Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T09:53:21.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex extreme measurable selections

Part of: Classical measure theory Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Douglas Mupasiri
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

MSC classification

Secondary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 2 , April 1995 , pp. 222 - 231
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Aumann, R. J., ‘Integrals of set-valued functions’, J. Math. Anal. Appl. 12 (1965), 1–12.CrossRefGoogle Scholar
[2]Cohn, D. L., Measure theory (Birkhäuser, Boston, 1980).CrossRefGoogle Scholar
[3]Davis, W. J., Garling, D. J. H. and Tomzcak-Jaegermann, N., ‘The complex convexity of quasinormed spaces’, J. Funct. Anal. 55 (1984), 110–150.CrossRefGoogle Scholar
[4]Dilworth, S. J., ‘Complex convexity and the geometry of Banach spaces’, Math. Proc. Cambridge Philos. Soc. 99 (1986), 495.CrossRefGoogle Scholar
[5]Dinculeanu, N., Vector measures, Internat. Ser. Monographs Pure Appl. Math. 95 (Pergamon Press, Oxford, 1967).CrossRefGoogle Scholar
[6]Greim, P., ‘An extremal vector-valued l p-function taking no extremal vectors as values’, Proc. Amer. Math. Soc. 84 (1982), 65–68.Google Scholar
[7]Johnson, J. A., ‘Extreme measurable selections’, Proc. Amer. Math. Soc. 44 (1974), 107–112.CrossRefGoogle Scholar
[8]Kuratowski, K. and Mostowski, A., Set theory, Stud. Logic Found. Math. (North-Holland, Amsterdam, 1976).Google Scholar
[9]Smith, M. A., Rotundity and extremity in l p(xi) and l p (μ, x), Contemp. Math. 52 (Amer. Math. Soc., Providence, 1980) pp. 143–162.Google Scholar
[10]Sundaresan, K., ‘Extreme points of the unit cell in Lebesgue—Bochner function spaces. I’, Proc. Amer. Math. Soc. 23 (1969), 179–184.Google Scholar
[11]Sundaresan, K., ‘Extreme points of the unit cell in Lebesgue—Bochner function spaces’, Colloq. Math. 22 (1970), 111–119.CrossRefGoogle Scholar
[12]Von Neumann, J., ‘On rings of operators, Reduction theory’, Ann. of Math. (2) 50 (1949), 401–485.CrossRefGoogle Scholar