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Vector Measures, Integration and Applications

  • Chapter
Positivity

Part of the book series: Trends in Mathematics ((TM))

Abstract

We will deal exclusively with the integration of scalar (i.e., ℝ or ℂ)-valued functions with respect to vector measures. The general theory can be found in [36, 37, 32], [44, Ch. I II] and [67, 124], for example. For applications beyond these texts we refer to [38, 66, 80, 102, 117] and the references therein, and the survey articles [33, 68]. Each of these references emphasizes its own preferences, as will be the case with this article. Our aim is to present some theoretical developments over the past 15 years or so (see §1) and to highlight some recent applications. Due to space limitation we restrict the applications to two topics. Namely, the extension of certain operators to their optimal domain (see §2) and aspects of spectral integration (see §3). The interaction between order and positivity with properties of the integration map of a vector measure (which is defined on a function space) will become apparent and plays a central role.

The first author acknowledges gratefully the support of D.G.I. # BFM2003-06335-C03-01 (Spain).

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  1. Facultad de Matemáticas, Universidad de Sevilla, Aptdo 1160, Sevilla, 41080, Spain

    G. P. Curbera

  2. Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072, Eichstätt, Germany

    W. J. Ricker

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  1. G. P. Curbera
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  2. W. J. Ricker
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Editors and Affiliations

  1. Institut Préparatoire aux Etudes Scientifiques et Techniques, Université de Carthage, B.P. 51, 2070, La Marsa, Tunisia

    Karim Boulabiar

  2. Department of Mathematics, University of Mississippi, Hume Hall 305, P. O. Box 1848, University, MS, 38677-1848, USA

    Gerard Buskes

  3. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060, Tunis, Tunisia

    Abdelmajid Triki

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© 2007 Birkhäuser Verlag AG2007

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Curbera, G.P., Ricker, W.J. (2007). Vector Measures, Integration and Applications. In: Boulabiar, K., Buskes, G., Triki, A. (eds) Positivity. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8478-4_5

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