Sublinear functionals and conical measures

  • Heinz KönigEmail author


The paper is devoted to the concept of conical measures which is central for the Choquet theory of integral representation in its final version. The conical measures need not be continuous under monotone pointwise convergence of sequences on the lattice subspace of functions which form their domain. We prove that they indeed become continuous (even in the nonsequential sense) when one restricts that domain to an obvious subcone. This result is in accord with the recent representation theory in measure and integration developed by the author. We also prove that one can pass from the subcone in question to a certain natural extended cone.


Linear Subspace Convex Cone Topological Vector Space Weak Topology Real Vector Space 
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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  1. [1]
    R. BECKER, Cônes Convexes en Analyse. Travaux en Cours 59, Paris 1999.Google Scholar
  2. [2]
    G. CHOQUET, Les cônes convexes faiblement complets dans l’analyse. In: Proc. Internat. Congr. Mathematicians (Stockholm 1962) 317–330. Inst. Mittag-Leffler 1963.Google Scholar
  3. [3]
    G. CHOQUET, Lectures on Analysis I–III. New York–Amsterdam 1969.Google Scholar
  4. [4]
    G. CHOQUET, Représentation intégrale. In: Measure Theory and its Applications (Sherbrooke 1982). LNM 1033, 114–143, Berlin–Heidelberg–New York 1983.Google Scholar
  5. [5]
    B. FUCHSSTEINER and H. KÖNIG, New versions of the Hahn-Banach theorem. In: Second Intern. Conf. General Inequalities (Oberwolfach 1978), ISNM 47, 255–266, Basel 1980.Google Scholar
  6. [6]
    B. FUCHSSTEINER and W. LUSKY, Convex Cones. Math. Stud. 56, Amsterdam–New York– Oxford 1981.Google Scholar
  7. [7]
    A. GROTHENDIECK, Espaces Vectoriels Topologiques. Soc. Mat. São Paulo 1964.Google Scholar
  8. [8]
    H. KÖNIG, Sublineare Funktionale. Arch. Math. 23, 500–508 (1972).Google Scholar
  9. [9]
    H. KÖNIG, On some basic theorems in convex analysis. In: Modern Applied Mathematics: Optimization and Operations Research (Summer School Bonn 1979) 107–144, Amsterdam–New York– Oxford 1982.Google Scholar
  10. [10]
    H. KÖNIG, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Berlin–Heidelberg–New York 1997.Google Scholar
  11. [11]
    H. KÖNIG, Measure and integration: Integral representations of isotone functionals. Annales Univ. Sarav. Ser. Math. 9, 123–153 (1998).Google Scholar
  12. [12]
    H. KÖNIG,Measure and integration: Comparison of old and new procedures. Arch. Math. 72, 192– 205 (1999).Google Scholar
  13. [13]
    H. KÖNIG, Review of [1]. Math. Reviews 2001 c: 46008.Google Scholar
  14. [14]
    H. KÖNIG and M. NEUMANN, Mathematische Wirtschaftstheorie, mit einer Einführung in die Konvexe Analysis. Math. Sys. Econom. 100, Hain-Athenäum 1986.Google Scholar
  15. [15]
    S. MAZUR and W. ORLICZ, Sur les éspaces métriques linéaires (II). Studia Math. 13, 137–179 (1953).MathSciNetCrossRefGoogle Scholar
  16. [16]
    R. PALLU DE LA BARRIÈRE, Intégration. Paris 1997.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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