Illinois Journal of Mathematics

The McShane and the Pettis integral of Banach space-valued functions defined on ${\Bbb R}\sp m$

Štefan Schwabik and Guoju Ye

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Abstract

In this paper, we define and study the McShane integral of functions mapping a compact interval $I_0$ in $R^m$ into a Banach space $X$. We compare this integral with the Pettis integral and prove, in particular, that the two integrals are equivalent if $X$ is reflexive and the unit ball of the dual $X^*$ satisfies an additional condition (P). This gives additional information on an implicitly stated open problem of R.A. Gordon and on the work of D.H. Fremlin and J. Mendoza.

Article information

Source
Illinois J. Math., Volume 46, Number 4 (2002), 1125-1144.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138470

Digital Object Identifier
doi:10.1215/ijm/1258138470

Mathematical Reviews number (MathSciNet)
MR1988254

Zentralblatt MATH identifier
1038.28009

Subjects
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
Secondary: 26A39: Denjoy and Perron integrals, other special integrals 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Citation

Ye, Guoju; Schwabik, Štefan. The McShane and the Pettis integral of Banach space-valued functions defined on ${\Bbb R}\sp m$. Illinois J. Math. 46 (2002), no. 4, 1125--1144. doi:10.1215/ijm/1258138470. https://projecteuclid.org/euclid.ijm/1258138470


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