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The McShane and the weak McShane integrals of Banach space-valued functions defined on $$\mathbb R^m$$. (English) Zbl 0993.28005
Summary: We define a concept of the weak McShane integral for functions mapping a compact interval $$I_0$$ in $$\mathbb{R}^m$$ into a Banach space $$X$$ and discuss the relation between the weak McShane integral and the Pettis integral. We show that the weak McShane integral and the Pettis integral are equivalent if and only if the Banach space $$X$$ contains no copy of $$c_0$$. Further, combining the properties of the McShane integral and Pettis integral, we get some equivalent statements concerning the McShane integral and the Pettis integral.

MSC:
 28B05 Vector-valued set functions, measures and integrals 26A39 Denjoy and Perron integrals, other special integrals 46G10 Vector-valued measures and integration