Real Analysis Exchange

Kurzweil-Henstock type integration on Banach spaces.

Luisa Di Piazza

Full-text: Open access

Abstract

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

Article information

Source
Real Anal. Exchange, Volume 29, Number 2 (2003), 543-555.

Dates
First available in Project Euclid: 7 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149698531

Mathematical Reviews number (MathSciNet)
MR2083796

Zentralblatt MATH identifier
1083.28007

Subjects
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Keywords
Kurzweil-Henstock integral Pettis integral equiintegrability

Citation

Di Piazza, Luisa. Kurzweil-Henstock type integration on Banach spaces. Real Anal. Exchange 29 (2003), no. 2, 543--555. https://projecteuclid.org/euclid.rae/1149698531


Export citation

References

  • B. Bongiorno, and L. Di Piazza, Convergence theorem for generalized Riemann-Stieltjes integrals, Real Anal. Exchange, 17 (1991-92), 339–361.
  • S. Cao, The Henstock integral for Banach-valued functions, SEA Bull. Math., 16 n. 1 (1992), 35–40.
  • J. Diestel, and J. J. Uhl, Vector measures, Math. Surveys, 15 (1977).
  • L. Di Piazza and V. Marraffa, The McShane, PU and Henstock integrals of Banach valued functions, Czechoslovak Math. J., 52 (127) n. 3 (2002), 609–633.
  • L. Di Piazza and K. Musial, Set valued Kurzweil-Henstock-Pettis integral, Set-valued Anal., to appear.
  • L. Di Piazza and D. Preiss, When do McShane and Pettis integrals coincide?, Ill. Math. Jour., 47 n. 4 (2003), 1177–1187.
  • D. H. Fremlin, Pointwise compact sets of measurable functions, Manuscripta Math., 15 (1975), 219–242.
  • D. H. Fremlin, The Henstock and McShane integrals of vector-valued functions, Ill. Math. Jour., 38 n. 3 (1994), 471–479.
  • D. H. Fremlin and J. Mendoza, On the integration of vector-valued functions, Ill. Math. Jour., 38 n.1 (1994), 127–147.
  • J. L. Gamez and J. Mendoza, On Denjoy-Dunford and Denjoy-Pettis integrals, Studia Math., 130 (1998), 115–133.
  • G. F. Geitz, Pettis integration, Proc. Amer. Math. Soc., 82 n. 1 (1981), 81–86.
  • R. Gordon, Another look at a convergence theorem for the Henstock integral, Real Anal. Exchange, 15 (1989/90), 724–728.
  • R. Gordon, The McShane integral of Banach-valued functions, Ill. Math. Jour., 34 n. 3 (1990), 557–567.
  • R. Gordon, Riemann integration in Banach spaces, Rocky Moun. J. Math., XXI 3, Summer 1991.
  • R. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Math., vol. 4, Am. Math. Soc. (1991).
  • R. C. James, Weak compactness and reflexivity, Israel J. Math., 2 (1964), 101–119.
  • J. Kurzweil, Nichtalsolut Konvergente Integral, Teubner, Leipzig (1980).
  • J. Kurzweil and J. Jarnik, Perron type integrable functions, Real Anal. Exchange, 17 (1991/92), 110–139.
  • J. Kurzweil and J. Jarnik, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exchange, 17 (1991/92), 110–139.
  • K. Musial, Topics in the theory of Pettis integration, Rendiconti Istit. Matem. Univ. Trieste, XXIII (1991), 177–262.
  • S. Swartz, Uniform integrability and mean convergence for the vector-valued McShane integral, Real Anal. Exchange, 23 (1) (1997/8), 303 –312.
  • S. Swartz, Norm convergence and uniform integrability for the Kurzweil-Henstock integral, Real Anal. Exchange, 24 (1) (1998/9), 423–426.
  • G. Ye, and S. Schwabik, The McShane and the Pettis integral of Banach space-valued functions defined on $\mathbb R^n$, Ill. Jour. Math., 46 n. 2 (2002), 1125–1144.
  • L. P. Yee, and R. Vyborny, The integral: An easy approach after Kurzweil and Henstock, Cambridge University Press (2000).