Positivity (2016) 20:877–916
Integrals for functions with values in a partially ordered
A. C. M. van Rooij
· W. B. van Zuijlen
Received: 22 May 2015 / Accepted: 7 December 2015 / Published online: 30 December 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract We consider integration of functions with values in a partially ordered vector
space, and two notions of extension of the space of integrable functions. Applying both
extensions to the space of real valued simple functions on a measure space leads to
the classical space of integrable functions.
Keywords Partially ordered vector space · Riesz space · Bochner integral ·
Pettis integral · Integral · Vertical extension · Lateral extension
Mathematics Subject Classiﬁcation 28B05 · 28B15
For functions with values in a Banach space there exist several notions of integration.
The best known are the Bochner and Pettis integrals (see [1,2]). These have been
thoroughly studied, yielding a substantial theory (see Chapter III in the book by Hille
and Phillips ).
As far as we know, there is no notion of integration for functions with values in a
partially ordered vector space; not necessarily a σ -Dedekind complete Riesz space.
In this paper we present such a notion. The basic idea is the following. (Here, E is a
partially ordered vector space in which our integrals take their values.)
W. B. van Zuijlen
Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen,
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands