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A note on measures with values in a partially ordered vector space

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Abstract

The goal of this note is to present an alternative, and we think simpler, proof of the following generalisation of the Riesz representation theorem due to J.D.M. Wright (Proc. London Math. Soc. 25 (1972) 675): any positive linear map Φ : C(X) →V can be represented by a V-valued measure on Baire subsets of X, where X is compact Hausdorff and V is a monotone σ-complete ordered vector space, not necessarily a lattice. Our proof suggests a purely inductive approach to measure theory, in the spirit of Borel’s original definition of measure of Borel sets.

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Authors and Affiliations

  1. Department of Computer Science, Chalmers University of Technology and Gotherburg University, SE-412 96, Göteborg, Sweden

    Thierry Coquand

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  1. Thierry Coquand
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Correspondence to Thierry Coquand.

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*Even for real valued measures, or in the case where V is a lattice, we believe that our approach gives essentially new proofs of basic results. Being purely inductive, it is an alternative to the use of outer measure, which, since Lebesgue’s work through Daniell, Caratheodory, Bourbaki, is the usual way to define the measure of Borel subsets. In particular, and in contrast to Wright’s work [7, 8], which relies for instance on the usual Riesz representation theorem, our proofs are developped independently of measure theory, and relies only on the inductive characterisation of the space of bounded Baire functions given in Lemma 1.1 and some general properties of ordered vector spaces.

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Coquand, T. A note on measures with values in a partially ordered vector space. Positivity 8, 395–400 (2004). https://doi.org/10.1007/s11117-004-7399-0

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