## Illinois Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.ijm/1258138470

Digital Object Identifier
doi:10.1215/ijm/1258138470

Mathematical Reviews number (MathSciNet)
MR1988254

Zentralblatt MATH identifier
1038.28009

#### 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