The McShane integral of Banach-valued functions. (English) Zbl 0685.28003

Gordon, Russell A.

Ill. J. Math. 34, No. 3, 557-567 (1990).

The McShane integral is an integral of Riemann-type that is equivalent to the Lebesgue integral. The mesh of the partition is controlled by a positive function rather than a constant and the tag of an interval need not belong to the interval. In this paper we consider the McShane integral of functions mapping a closed interval into a real Banach space. The main result is that every measurable, Pettis integrable function is McShane integrable. These two integrals are equivalent in separable spaces that do not contain a copy of \(c_ 0\).

Reviewer: R.A.Gordon

MSC:

28B05	Vector-valued set functions, measures and integrals
26A42	Integrals of Riemann, Stieltjes and Lebesgue type

Keywords:

Banach-valued; integral of Riemann-type; McShane integral of functions mapping a closed interval into a real Banach space.; Pettis integrable function