Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T21:07:46.465Z Has data issue: false hasContentIssue false
  • English
  • Français

A CHARACTERIZATION OF THE WEAK RADON–NIKODÝM PROPERTY BY FINITELY ADDITIVE INTERVAL FUNCTIONS

Part of: Measures, integration, derivative, holomorphy Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  07 September 2009

B. BONGIORNO ,
L. DI PIAZZA  and
K. MUSIAŁ
B. BONGIORNO
Affiliation:
Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy (email: bbongi@math.unipa.it)
L. DI PIAZZA
Affiliation:
Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy (email: dipiazza@math.unipa.it)
K. MUSIAŁ*
Affiliation:
Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email: musial@math.uni.wroc.pl)
*
✉For correspondence; e-mail: musial@math.uni.wroc.pl
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.

A characterization of Banach spaces possessing the weak Radon–Nikodým property is given in terms of finitely additive interval functions. Due to that characterization several Banach space valued set functions that are only finitely additive can be represented as integrals.

Keywords

Kurzweil–Henstock integralPettis integralvariational measureweak Radon–Nikodým property

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46G05: Derivatives 46G10: Vector-valued measures and integration
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 476 - 485
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

The first and second authors were partially supported by MiUR, and all the authors were partially supported by grant N. 201 00932/0243.

References

[1]Diestel, J. and Uhl, J. J., Vector Measures, Mathematical Surveys, 15 (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
[2]Di Piazza, L., ‘Variational measures in the theory of the integration in ℝm’, Czechoslovak Math. J. 51 (2001), 95–110.CrossRefGoogle Scholar
[3]Di Piazza, L. and Musiał, K., ‘Characterizations of Kurzweil–Henstock–Pettis integrable functions’, Studia Math. 176 (2006), 159–176.CrossRefGoogle Scholar
[4]Musiał, K., ‘A characterization of the weak Radon–Nikodym property in terms of the Lebesgue measure’, in: Proceedings of the Conference Topology and Measure III, Part 1, 2 (Vitte/Hiddensee, 1980) (Wissensch. Beitr., Ernst-Moritz-Arndt Univ., Greifswald, 1982), pp. 163–168.Google Scholar
[5]Musiał, K., ‘Topics in the theory of Pettis integration’, Rend. Instit. Mat. Univ. Trieste 23 (1991), 177–262.Google Scholar
[6]Musiał, K., ‘Pettis integral’, in: Handbook of Measure Theory I (ed. E. Pap) (Elsevier, Amsterdam, 2002), pp. 531–586.Google Scholar
[7]Pfeffer, W. F., ‘The Lebesgue and Denjoy–Perron integrals from a descriptive point of view’, Ric. Mat. 48 (1999), 211–223.Google Scholar
[8]Talagrand, M., ‘Pettis integral and measure theory’, Mem. Amer. Math. Soc. 51(307) (1984).Google Scholar
[9]Thomson, B. S., ‘Derivatives of interval functions’, Mem. Amer. Math. Soc. 452 (1991).Google Scholar