In this paper, we define the Dunford–Henstock–Kurzweil and the Dunford–McShane integrals of Banach space-valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space \({\mathbb {R}}^{m}\). We will show that the new integrals are “natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock–Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.

中文翻译：

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在m维有界集上定义的向量值函数的Dunford–Henstock–Kurzweil和Dunford–McShane积分

在本文中，我们定义了在m维欧式空间\（{{mathbb {R}} ^ {m}的有界Lebesgue可测子集上定义的Banach空间值函数的Dunford–Henstock–Kurzweil和Dunford–McShane积分\）。我们将证明新积分是McShane和Henstock-Kurzweil积分的“自然”扩展，它从m维封闭的非退化区间扩展到m维有界Lebesgue可测集。作为应用程序，我们将根据积分对McShane和Henstock-Kurzweil积分进行完整的描述。而且，新积分之间的关​​系将根据邓福德积分来证明。