Real Analysis Exchange

Strong convergence in Henstock-Kurzweil-Pettis integration under an extreme point condition.

B. Satco

Full-text: Open access

Abstract

In the present paper, some Olech and Visintin-type results are obtained in Henstock-Kurzweil-Pettis integration. More precisely, under extreme or denting point condition, one can pass from weak convergence (i.e. convergence with respect to the topology induced by the tensor product of the space of real functions of bounded variation and the topological dual of the initial Banach space) or from the convergence of integrals to strong convergence (i.e. in the topology of Alexiewicz norm or, even more, of Pettis norm). Our results extend the results already known in the Bochner and Pettis integrability setting.

Article information

Source
Real Anal. Exchange Volume 31, Number 1 (2005), 179-194.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516809

Mathematical Reviews number (MathSciNet)
MR2218197

Zentralblatt MATH identifier
1111.28009

Subjects
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] 26A39: Denjoy and Perron integrals, other special integrals 46B20: Geometry and structure of normed linear spaces

Keywords
Keywords: Henstock-Kurzweil-Pettis integral extreme point denting point

Citation

Satco, B. Strong convergence in Henstock-Kurzweil-Pettis integration under an extreme point condition. Real Anal. Exchange 31 (2005), no. 1, 179--194.https://projecteuclid.org/euclid.rae/1149516809


Export citation

References

  • A. Amrani, C. Castaing, Weak compactness in Pettis integration, Bull. of the Polish Academy of Sciences Mathematics, 45, no. 2 (1997), 139–150.
  • A. Amrani, C. Castaing, M. Valadier, Convergence in Pettis Norm Under Extreme Point Condition, Vietnam J. Math., 26, no. 4 (1998), 323–335.
  • E. J. Balder, C. Hess, Two generalizations of Komlós theorem with lower closure-type applications, J. Convex Anal., 3 (1996), 25–44.
  • E. J. Balder, A. R. Sambucini, A note on strong convergence for Pettis integrable functions, Vietnam J. Math., 31, no. 3 (2003), 341–347.
  • A. Bourras, C. Castaing, M. Gouessous, Olech-types lemma and Visintin-types theorem in Pettis integration and $L_{E^{^{\prime }}}^{1}[E]$, NLA98: Convex analysis and chaos (Sakado, 1998), 1–26, Josai Math. Monogr., 1, Josai Univ., Sakado, 1999.
  • C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.
  • S. S. Cao, The Henstock integral for Banach-valued functions, SEA Bull. Math., 16 (1992), 35–40.
  • L. Di Piazza, Kurzweil-Henstock type integration on Banach spaces, Real Anal. Exchange, 29(2) (2003/2004), 543–556.
  • L. Di Piazza, K. Musial, Set-Valued Kurzweil-Henstock-Pettis Integral, Set-Valued Analysis, 13, 2 (2005), 167–179.
  • K. El Amri, C. Hess, On the Pettis Integral of Closed Valued Multifunctions, Set-Valued Analysis, 8 (2000), 329–360.
  • R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. in Math., 4, 1994.
  • K. Musial, Topics in the theory of Pettis integration, in School of Measure theory and Real Analysis, Grado, Italy, May 1992.