New versions of the Daniell-Stone-Riesz representation theorem

  • Heinz KönigEmail author


The traditional representation theorems after Daniell-Stone and Riesz were in a kind of separate existence until Pollard-Topsøe 1975 and Topsøe 1976 were the first to put them under common roofs. In the same spirit the present article wants to obtain a unified representation theorem in the context of the author’s work in measure and integration. It is an inner theorem like the previous ones. The basis is the recent comprehensive inner Daniell-Stone theorem, so that in particular there are no a priori assumptions on the additive behaviour of the data.


Daniell-Stone and Riesz representation theorems Stonean lattices and Stonean lattice cones Inner premeasures Inner extensions Inner envelopes Choquet integral Set-theoretical compactness 


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  1. 1.
    B. Anger, C. Portenier, Radon integrals, Progress in Math. vol. 103, Birkhäuser (1992).Google Scholar
  2. 2.
    B. Anger, C. Portenier, Radon integrals and Riesz representation, Rend. Circ. Mat. Palermo (2) 28 (1992) Suppl. 269–300.Google Scholar
  3. 3.
    E. Behrends, Mass- und Integrationstheorie, Springer (1987).Google Scholar
  4. 4.
    C. Berg, J.R.P. Christensen, P. Ressel, Harmonic Analysis on Semigroups, Grad. Texts Math. vol. 100, Springer (1984).Google Scholar
  5. 5.
    N. Bourbaki, Intégration, Chap.IX, Hermann (1969). English translation, Springer (2004).Google Scholar
  6. 6.
    J. Elstrodt, Mass- und Integrationstheorie, 4th ed., Springer (2005).Google Scholar
  7. 7.
    D.H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge Univ. Press (1974).CrossRefGoogle Scholar
  8. 8.
    D.H. Fremlin, Measure theory, vol. 4. Torres Fremlin (2003). "".
  9. 9.
    J. Kisyński, On the generation of tight measures, Studia Math. 30 (1968), 141–151.MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. König, Measure and Integration: An Advanced Course in Basic Procedures and Applications, Springer (1997).Google Scholar
  11. 11.
    H. König, Measure and Integration: Integral representations of isotone functionals, Annales Univ. Saraviensis Ser. Math. 9(2) (1998), 123–153.Google Scholar
  12. 12.
    H. König, Measure and Integration: An attempt at unified systematization, Rend. Istit. Mat. Univ. Trieste 34 (2002), 155–214.Google Scholar
  13. 13.
    H. König, The (sub/super)additivity assertion of Choquet, Studia Math. 157 (2003), 171–197.MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. König, Projective limits via inner premeasures and the true Wiener measure, Mediterranean J. Math. 1 (2004), 3–42.MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. König, Measure and integral: new foundations after one hundred years, In: Proc. Conf. Evolution Equations, U Mons (Belgium) and U Valenciennes (France) 28 August–1 September (2006).Google Scholar
  16. 16.
    D. Pollard, F. Topsøe, A unified approach to Riesz type representation theorems, Studia Math. 54 (1975), 173–190.MathSciNetCrossRefGoogle Scholar
  17. 17.
    L.A. Steen, J.A. Seebach, Jr., Counterexamples in Topology, 2nd ed., Springer (1978).Google Scholar
  18. 18.
    F. Topsøe, Compactness in spaces of measures, Studia Math. 36 (1970), 195–212.MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. Topsøe, Topology and measure, Lect. Notes Math. vol. 133, Springer (1970).Google Scholar
  20. 20.
    F. Topsøe, Further results on integral representations, Studia Math. 55 (1976), 239–245.MathSciNetCrossRefGoogle Scholar
  21. 21.
    F. Topsøe, Radon measures, some basic constructions, In: Measure Theory and its Applications (Sherbrooke 1982), Lecture Notes Math., 1033, pp. 303–311. Springer (1983).Google Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany

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