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Convergence and comparison results for Henstock-Kurzweil and McShane integrable vector-valued functions. (English) Zbl 1240.26025
Summary: A monotone convergence theorem is proved for Henstock-Kurzweil and McShane integrable functions from a compact real interval \([a,b]\) to an ordered Banach space \(X\) with a regular order cone \(X_+\). As an application we show that, if \(X\) is a weakly sequentially complete Banach space and \(X_+\) is normal order cone, then \(f:[a,b]\to X_+\) is Henstock-Kurzweil integrable if and only if \(f\) is McShane integrable. If \(f\) is strongly Henstock-Kurzweil integrable, we prove that \(f\) is McShane integrable without the weak completeness hypothesis on \(X\).

MSC:
26B12 Calculus of vector functions
28B15 Set functions, measures and integrals with values in ordered spaces
46G10 Vector-valued measures and integration
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