Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T07:50:36.815Z Has data issue: false hasContentIssue false
  • English
  • Français

APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS

Published online by Cambridge University Press:  01 September 2008

B. BONGIORNO ,
L. DI PIAZZA  and
K. MUSIAŁ
B. BONGIORNO
Affiliation:
Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy e-mail: bbongi@math.unipa.it
L. DI PIAZZA
Affiliation:
Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy e-mail: dipiazza@math.unipa.it
K. MUSIAŁ
Affiliation:
Wrocław University, Institute of Mathematics, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland 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.

The approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.

Keywords

Primary 28B20Secondary 26A3928B0546G1054C60
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 583 - 593
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Alexiewicz, A., Linear functionals on Denjoy integrable functions, Coll. Math. 1 (1948), 289–293.CrossRefGoogle Scholar
2.Di Piazza, L., Weak convergence of Henstock integrable sequence, J. Math. Study 27 (1994), 148–153.Google Scholar
3.Di, L.Piazza, Kurzweil–Henstock type integration on Banach space, Real Anal. Ex. 29 (2) (2003–04), 543–556.Google Scholar
4.Di Piazza, L. and Musiał, K., Set valued Kurzweil–Henstock–Pettis integral, Set-Valued Anal. 13 (2005), 165–179.CrossRefGoogle Scholar
5.Di Piazza, L. and Musiał, K., Characterizations of Kurzweil–Henstock–Pettis integrable functions, Studia Math. 176 (2) (2006), 159–176.CrossRefGoogle Scholar
6.Di Piazza, L. and Musiał, K., A decomposition theorem for compact-valued Henstock Integral, Monatsch. Math. 148 (2006), 119–126.CrossRefGoogle Scholar
7.Diestel, J. and Uhl, J. J., Vector measures, Math. Surveys 15 (1977).Google Scholar
8.Ene, V., The wide Denjoy integral as the limit of a sequence of step functions in a suitable convergence, Real Anal. Ex. 23 (2) (1997/98), 719–734.CrossRefGoogle Scholar
9.Fremlin, D. H. J. and Talagrand, M., A decomposition theorem for additive set functions and applications, Math. Z. 168 (1979), 117–142.CrossRefGoogle Scholar
10.Gamez, J. L. and Mendoza, J., On Denjoy–Dunford and Denjoy–Khintchine–Pettis integrals, Studia Math. 130 (1998), 115–133.Google Scholar
11.Gordon, R. A., The Denjoy extension of the Bochner, Pettis and Dunford integrals, Studia Math. 92 (1989), 73–91.CrossRefGoogle Scholar
12.Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Grad. Stud. Math. 4 (1994).CrossRefGoogle Scholar
13.Musiał, K., Martingales of Pettis integrable functions, Lecture Notes in Math. 794 (1980), 324–339.Google Scholar
14.Musiał, K., Pettis integration, Suppl. Rend. Circolo Mat. di Palermo, Ser II, 10 (1985), 133–142.Google Scholar
15.Musiał, K., Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste 23 (1991), 177–262.Google Scholar
16.Musiał, K., Pettis Integral, Handbook of measure theory I, 532–586. (Elsevier Science B. V. Amsterdam, 2002).Google Scholar
17.Saks, S., Theory of the integral, 2nd revised ed. (Hafner, New York, 1937).Google Scholar
18.Swartz, Ch., Norm convergence and uniform integrability for the Henstock–Kurzweil integral, Real Anal. Ex. 24 (1998/99), 423–426.CrossRefGoogle Scholar
19.Talagrand, M., Pettis integral and measure theory, Memoirs AMS 307 (1984).Google Scholar