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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
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Abstract

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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

Information

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

References

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