×

zbMATH — the first resource for mathematics

Approximation of Banach space valued non-absolutely integrable functions by step functions. (English) Zbl 1154.28006
The authors study the problem of approximating Banach space valued integrable functions by a sequence of step functions. The authors consider generalizations of the Henstock-Kurzweil and Denjoy-Khintchine integrals to vector valued functions. The generalization of the Henstock-Kurzweil (HK) integral to Banach space valued functions is straightforward and the authors show that an HK integrable function can always be approximated by a sequence of step functions in the Alexiwiecz norm. If \(X\) is a Banach space, a function \(f:[0,1]\rightarrow X\) is scalarly HK integrable if \(x'f\) is HK integrable for every \(x'\in X'\) and is HK-Pettis (HKP) integrable if \(f\) is scalarly HK integrable and for every subinterval \(I\subset [0,1]\) there exists \(w_I\in X\) such that \(x'w_I=HK\int_Ix'f\) for every \(x'\in X'\); set \(w_{I}=\int_{I}f\). The authors show that an HKP integrable function \(f\) can be approximated by a sequence of step functions in the Alexiewicz norm iff the range of the indefinite integral, \(\{\int_{I}f:I\) a subinterval of \([0,1]\}\), is relatively norm compact. Several other interesting results for these integrals are also established.
The authors consider analogous results for vector valued generalizations of the Denjoy-Khintchine integral. This paper is a continuation of previous papers by the authors and draw on the results of these papers.

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
26A39 Denjoy and Perron integrals, other special integrals
28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
54C60 Set-valued maps in general topology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Talagrand, Memoirs AMS 307 pp none– (1984)
[2] Alexiewicz, Coll. Math. 1 pp 289– (1948)
[3] Saks, Theory of the integral (1937)
[4] Musia?, Handbook of measure theory I pp 532– (2002)
[5] Musia?, Rend. Istit. Mat. Univ. Trieste 23 pp 177– (1991)
[6] DOI: 10.1007/BF01214191 · Zbl 0393.28005 · doi:10.1007/BF01214191
[7] Musia?, Suppl. Rend. Circolo Mat. di Palermo, Ser II 10 pp 133– (1985)
[8] Ene, Real Anal. Ex. 23 pp 719– (1997)
[9] Musia?, Martingales of Pettis integrable functions 794 pp 324– (1980)
[10] Diestel, Math. Surveys 15 pp none– (1977)
[11] Gordon, Grad. Stud. Math. 4 pp none– (1994)
[12] DOI: 10.1007/s00605-005-0376-2 · Zbl 1152.28016 · doi:10.1007/s00605-005-0376-2
[13] Gordon, Studia Math. 92 pp 73– (1989)
[14] Di Piazza, Studia Math. 176 pp 159– (2006)
[15] Gamez, Studia Math. 130 pp 115– (1998)
[16] DOI: 10.1007/s11228-004-0934-0 · Zbl 1100.28008 · doi:10.1007/s11228-004-0934-0
[17] Di, Real Anal. Ex. 29 pp 543– (2003)
[18] Di Piazza, J. Math. Study 27 pp 148– (1994)
[19] Swartz, Real Anal. Ex. 24 pp 423– (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.