## Illinois Journal of Mathematics

### A variational Henstock integral characterization of the Radon–Nikodým property

#### Abstract

A characterization of Banach spaces possessing the Radon–Nikodým property is given in terms of finitely additive interval functions. We prove that a Banach space $X$ has the RNP if and only if each $X$-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an $X$-valued function. Due to that characterization several $X$-valued set functions that are only finitely additive can be represented as integrals.

#### Article information

Source
Illinois J. Math., Volume 53, Number 1 (2009), 87-99.

Dates
First available in Project Euclid: 22 January 2010

https://projecteuclid.org/euclid.ijm/1264170840

Digital Object Identifier
doi:10.1215/ijm/1264170840

Mathematical Reviews number (MathSciNet)
MR2584936

Zentralblatt MATH identifier
1200.46021

#### Citation

Bongiorno, B.; Di Piazza, L.; Musiał, K. A variational Henstock integral characterization of the Radon–Nikodým property. Illinois J. Math. 53 (2009), no. 1, 87--99. doi:10.1215/ijm/1264170840. https://projecteuclid.org/euclid.ijm/1264170840

#### References

• B. Bongiorno, L. Di Piazza and V. Skvortsov, A new full descriptive characterization of Denjoy-Perron integral, Real Analysis Exchange 21 (1995/96), 256–263.
• D. Bongiorno, Stepanoff's theorem in separable Banach spaces, Comment. Math. Univ. Carolinae 39 (1998), 323–335.
• S. S. Cao, The Henstock integral for Banach-valued functions, SEA Bull. Math. 16 (1992), 35–40.
• J. Diestel and J. J. Uhl, Vector measures, Math. Surveys, vol. 15, Amer. Math. Soc., Providence, RI, 1977.
• S. J. Dilworth and M. Girardi, Nowhere weak differentiability of the Pettis integral, Quest. Math. 18 (1995), 365–380.
• L. Di Piazza, Varational measures in the theory of the integration in $R^m$, Czechoslovak. Math. J. 51 (2001), 95–110.
• L. Di Piazza and V. Marraffa, The McShane, PU and Henstock integrals of Banach valued functions, Czechoslovak Math. J. 52 (2002), 609–633.
• R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Math., vol. 4, Amer. Math. Soc., Providence, RI, 1994.
• V. Marraffa, A descriptive characterization of the variational Henstock integral, Proceedings of the International Mathematics Conference (Manila, 1998), Matimyás Mat. 22 (1999), 73–84.
• K. Musiał, Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste 23 (1991), 177–262.
• ––––, Pettis integral, Handbook of Measure Theory I, Elsevier, Amsterdam, 2002, pp. 531–586.
• W. F. Pfeffer, The Lebesgue and Denjoy-Perron integrals from a descriptive point of view, Ricerche di Mat. XLVIII (1999), 211–223.
• S. Schwabik and Y. Guoju, Topics in Banach space integration, Series in Real Analysis, vol. 10, World Scientific, Hackensack, NJ, 2005.
• V. A. Skvortsov and A. P. Solodov, A descriptive characterization of the Denjoy-Bochner integral and its generalizations, Moscow University Mathematics Bulletin 57 (2002), 36–39.
• A. P. Solodov, Riemann-type definition for the restricted Denjoy-Bochner integral, Fundamentalnaya i Prikladnaya Matematika\goodbreak 7 (2001), 887–895.
• B. Thomson, Derivatives of interval functions, Mem. Amer. Math. Soc. 452 (1991).