, Volume 20, Issue 1, pp 209–233 | Cite as

Dual representation of convex sets of probability measures on totally bounded spaces

  • Robert SchlichtEmail author


Convex sets of probability measures, frequently encountered in probability theory and statistics, can be transparently analyzed by means of dual representations in a function space. This paper introduces totally bounded spaces, whose structure is defined by a set of bounded real-valued functions, as a general framework for studying such representations. The reinterpretation of classical theorems in this framework clarifies the role of compactness and leads to simple existence criteria. Applications include results on the existence of probability measures satisfying given sets of conditions and an equivalence of consistent preferences and families of probability measures. Moreover, countable additivity of probabilities is seen to be a consequence of elementary consistency assumptions.


Probability measure Duality Convex cone Decision theory Uniform space 

Mathematics Subject Classification

60B05 62C05 46A55 54E15 



I would like to thank a number of colleagues for valuable discussions and, moreover, the referee for a very careful reading of the manuscript, which led to considerable improvements.


  1. 1.
    Aliprantis, C.D., Tourky, R.: Cones and Duality. American Mathematical Society, Providence (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bernoulli, D.: Specimen theoriae novae de mensura sortis. Commun. Acad. Petrop. Vol. V, pp. 195–192 (1738). Transl. in. Econometrica 22, 23–36 (1954)Google Scholar
  3. 3.
    Bernoulli, J.: Latin version of C. Huygens, Van rekeningh in spelen van geluck. In Ars conjectandi, p. 3, Basel (1713)Google Scholar
  4. 4.
    Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer, New York (1980)CrossRefGoogle Scholar
  5. 5.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)zbMATHGoogle Scholar
  6. 6.
    Bogachev, V.I.: Measure Theory, vol. II. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bourbaki, N.: Topological Vector Spaces (transl. from French). Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bourbaki, N.: General Topology, Chapters 1–4 (transl. from French). Springer, Berlin (1989)zbMATHGoogle Scholar
  9. 9.
    Cartier, P., Fell, J.M.G., Meyer, P.-A.: Comparaison des mesures portées par un ensemble convexe compact. Bull. Soc. Math. Fr. 92, 435–445 (1964)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Choquet, G.: Lectures on Analysis, vol. II. W.A. Benjamin Inc, London (1969)Google Scholar
  11. 11.
    De Finetti, B.: La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut Henri Poincaré 7, 1–68 (1937). Transl. in Kotz, S., Johnson, N.L. (eds.), Breakthroughs in statistics, vol. I, pp. 134–174. Springer, New York (1991)Google Scholar
  12. 12.
    Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fremlin, D.H.: Topological Riesz Spaces. Cambridge University Press, Cambridge (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fremlin, D.H.: Measure Theory, vol. 4. Torres Fremlin, Colchester (2003)zbMATHGoogle Scholar
  15. 15.
    Fuchssteiner, B., Lusky, W.: Convex Cones. North-Holland Pub. Co., Amsterdam (1981)zbMATHGoogle Scholar
  16. 16.
    Glicksberg, I.: The representation of functionals by integrals. Duke Math. J. 19, 253–261 (1952)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kamae, T., Krengel, U., O’Brien, G.L.: Stochastic inequalities on partially ordered spaces. Ann. Probab. 5, 899–912 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Kelley, J.L.: General Topology. Van Nostrand, New York (1955)zbMATHGoogle Scholar
  20. 20.
    Kolmogoroff, A.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933)CrossRefGoogle Scholar
  21. 21.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)zbMATHGoogle Scholar
  22. 22.
    Nelson, E.: Regular probability measures on function space. Ann. Math. 2(69), 630–643 (1959)CrossRefGoogle Scholar
  23. 23.
    Neveu, J.: Mathematical foundations of the calculus of probability (transl. from French). Holden-Day Inc, San Francisco (1965)Google Scholar
  24. 24.
    Pachl, J.: Uniform Spaces and Measures. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  25. 25.
    Ramsey, F.P.: Truth and probability (1926). In: Ramsey, F.P., Braithwaite, R.D. (eds.) The Foundations of Mathematics, pp. 156–198. Paul, Trench, Trubner, London (1931)Google Scholar
  26. 26.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  27. 27.
    Savage, L.J.: The Foundations of Statistics, 2nd edn. Dover Publications, New York (1972)zbMATHGoogle Scholar
  28. 28.
    Schaefer, H.H.: Topological Vector Spaces 2nd ed. with M. P. Wolf. Springer, New York (1999)CrossRefGoogle Scholar
  29. 29.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Varadarajan, V.S.: Measures on topological spaces, transl. from Russian. Am. Math. Soc. Transl. 48, 161–228 (1965)Google Scholar
  31. 31.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 3rd edn. Princeton University Press, Princeton (1953)zbMATHGoogle Scholar
  32. 32.
    Wald, A.: Statistical Decision Functions. Wiley, New York (1950)zbMATHGoogle Scholar
  33. 33.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefzbMATHGoogle Scholar
  34. 34.
    Zaanen, A.C.: Riesz Spaces II. North-Holland Pub. Co., Amsterdam (1983)zbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of Environmental SciencesTU DresdenTharandtGermany

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