Abstract
We prove some convergence theorems for the Henstock-Kurzweil- Pettis and Denjoy-Pettis integrals. Since these integrals are more general than some “classical” non-absolute integrals and than the Pettis integral, we generalize well-known convergence theorems for both types of the mentioned integrals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. S. Cao, The Henstock integral for Banach valued functions, SEA Bull. Math., 16 (1992), 36–40.
V. Celidze and A. G. Dzhvarsheishvili, Theory of Denjoy Integral and some of its Applications (Tbilisi, 1987) (in Russian).
M. Cichoń, I. Kubiaczyk and A. Sikorska, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, to appear.
J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15 (Providence, Rhode Island, (1977).
J. L. Gámez and J. Mendoza, On Denjoy-Dunford and Denjoy-Pettis integrals, Studia Math., 130 (1998), 115–133.
R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc., 82 (1981), 81–86.
R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock (Providence, Rhode Island, 1994).
R. A. Gordon, The Denjoy extension of the Bochner, Pettis, and Dunford integrals, Studia Math., 92 (1989), 73–91.
R. Henstock, The General Theory of Integration, Oxford Mathematical Monographs, Clarendon Press (Oxford, 1991).
R. C. James, Weak compactness and reflexivity, Israel J. Math, 2 (1964), 101–119.
J. Kurzweil, Nichtabsolut Konvergente Integrale, Teubner Texte zür Mathematik No 26 (Leipzig, 1980).
K. Musiał, Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces, Alli Sem. Mat. Fis. Univ. Modena, 35 (1987), 159–166.
K. Musiał, Topics in the theory of Pettis integration, Rend. Instit. Mat. Univ. Trieste, 23 (1993), 177–262.
B. K. Pal and S. N. Mukhopadhyay, The Cesàro-Denjoy-Pettis scale of integration, Acta Math. Hungar., 45 (1985), 289–295.
J. M. Park and D. H. Lee, The Denjoy extension of the McShane integral, Bull. Korean Math. Soc., 33 (1996), 411–417.
L. P. Yee, Lanzhou Lectures on Henstock Integration, World Scientific (Singapore, (1989).
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc., 44 (1938), 277–304.
S. Schwabik and I. Vrkoč, On Kurzweil-Henstock equiintegrable sequences, Math. Bohem., 121 (1996), 189–207.
Ch. Swartz, Norm convergence and uniform integrability for the Henstock-Kurzweil integral, Real Anal. Exch., 24 (1998/99), 423–426.
L. P. Yee and Ch. T. Seng, A better convergence theorem for Henstock integrals, Bull. London Math. Soc., 17 (1987), 557–564.
L. P. Yee and Ch. T. Seng, A short proof of the controlled convergence theorem for Henstock integrals, Bull. London Math. Soc., 19 (1987), 60–62.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cichon, M. Convergence Theorems for the Henstock—Kurzweil—Pettis Integral. Acta Mathematica Hungarica 92, 75–82 (2001). https://doi.org/10.1023/A:1013756111769
Issue Date:
DOI: https://doi.org/10.1023/A:1013756111769