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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.

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
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