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 \(X\).


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