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Description of the limit set of Henstock-Kurzweil integral sums of vector-valued functions. (English) Zbl 1309.46024

Let \(X\) be a separable Banach space and \(f:[0,1]\to X\) be an \(i\)-bounded function. The following statement is a synthesis of the main results in the paper.
Theorem. Under these conditions, we have
(i)
\(I_{HK}(f)=V(f)\); hence, \(I_{HK}(f)\) is nonempty convex;
(ii)
there exists a multifunction \(Q:[0,1]\to \text{cl}(X)\) such that \(I_{HK}(f|_J)=\overline{(B)\int_J Qd\lambda}\) for each interval \(J:=[a,b]\) of \([0,1]\).
Some other aspects involving the set \(I_{HK}(f)\) are also discussed.

MSC:

46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
26E20 Calculus of functions taking values in infinite-dimensional spaces
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