Convergence and comparison results for Henstock-Kurzweil and McShane integrable vector-valued functions. (English) Zbl 1240.26025

Heikkilä, S.; Ye, Guoju

Southeast Asian Bull. Math. 35, No. 3, 407-418 (2011).

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

Keywords:

Henstock-Kurzweil integrability; McShane integrability; ordered Banach space; monotone convergence theorem