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Positivity

, Volume 8, Issue 4, pp 395–400 | Cite as

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

  • Thierry Coquand
Article

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.

Keywords

Vector Space Fourier Analysis Operator Theory London Math Potential Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Borel E. Leçons sur la Théorie des Fonctions. Gauthier-Villars, quatrième édition 1950, 1ere édition 1898Google Scholar
  2. 2.
    Lusin N. Leçons sur les ensembles analytiques. Gauthier-Villars, 1930Google Scholar
  3. 3.
    Luxemburg W.A., Zaanen A.C. Riesz Spaces I. North-Holland, 1971Google Scholar
  4. 4.
    Riesz, M. 1930Sur la décomposition des opérations fonctionelles linéairesAtti del Congr. Internaz. dei Mat. Bologna 19283143148Google Scholar
  5. 5.
    Stone, M.H. 1941A general theory ofspectra IIProc. Natl. Acad. Sci. USA.278387Google Scholar
  6. 6.
    Stone, M.H. 1949Boundedness properties in function-latticesCanadian Journal of Mathematics.1177186Google Scholar
  7. 7.
    Wright, J.D.M. 1969Stone-algebra-valued measures and integralsProc. London Math. Soc.19107122Google Scholar
  8. 8.
    Wright, J.D.M. 1972Measures with values in a partially ordered vector spaceProc. London Math. Soc.25675688Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of Computer ScienceChalmers University of Technology and Gotherburg UniversityGöteborgSweden

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