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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