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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.

##### 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
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