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On open \(\left( c,\epsilon \right) \)-balls in topological spaces that capture convergence in non-additive probability measure with probability-one coincidence


We introduce the topological spaces \(\left( L,\tau _{\nu 1}\right) \) which are well-defined for any given subset of random variables L on any given non-additive probability space \((\Omega ,\mathcal {F},\nu )\). A \(\left( c,\epsilon \right) \)-ball at \(X\in L\) contains all random variables in L that are sufficiently close to X in the sense that any payoff differences to X smaller than \(c>0\) happen with \(\nu \)-probability greater than \(1-\epsilon \). We derive two main results concerning \(\left( c,\epsilon \right) \)-balls. Firstly, all \(\left( c,\epsilon \right) \)-balls must be open sets in \(\left( L,\tau _{\nu 1}\right) \) whenever \(\nu \) is continuous from below and dual-autocontinuous from above. In that case, convergence of sequences of random variables on \(\left( L,\tau _{\nu 1}\right) \) is equivalent to convergence in non-additive probability measure \(\nu \) with probability-one coincidence. Secondly, an open \(\left( c,\epsilon \right) \)-ball cannot be a convex strict subset of L whenever L has a non-trivial local cone structure and \((\Omega ,\mathcal {F},\nu )\) is dual-nonatomic.

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  1. For formal definitions of such structural properties see, e.g., Denneberg [8, Chapter 2] Additive probability measures are non-additive probability measures that are both concave and convex.

  2. For the special case of subjective additive beliefs given as the Lebesgue measure \(\lambda \) defined on the open unit interval, Assa and Zimper [3] and Zimper and Assa [23] discuss utility- and/or risk-measure representations of preferences over random variables that are (semi-)continuous in the topology of convergence in \(\lambda \).

  3. Compare., e.g., Theorem 13.41(3) in Aliprantis and Border [1], Paragraph 1.47 in Rudin [14], Theorem 1 in Day [7].


  1. Aliprantis, D.C., Border, K.: Infinite Dimensional Analysis, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Arhangel’skiĭ, A.V.: Mappings and spaces. Russ. Math. Surv. 21, 115–162 (1966)

    Article  Google Scholar 

  3. Assa, H., Zimper, A.: Preferences over all random variables: incompatibility of convexity and continuity. J. Math. Econ. 75, 71–83 (2018)

    MathSciNet  Article  Google Scholar 

  4. Billingsley, P.: Probability and measure. Wiley, New York (1995)

    MATH  Google Scholar 

  5. Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals III: topology determined by the integral. Fuzzy Sets Syst. 304, 20–34 (2016)

    MathSciNet  Article  Google Scholar 

  6. Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1954)

    MathSciNet  Article  Google Scholar 

  7. Day, M.M.: The Spaces \(L^{p}\) with \(0<p<1\). Bull. Am. Math. Soc. 46, 816–823 (1940)

    Article  Google Scholar 

  8. Denneberg, D.: Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  9. Ellsberg, D.: Risk, ambiguity and the savage axioms. Quart. J. Econ. 75, 643–669 (1961)

    MathSciNet  Article  Google Scholar 

  10. Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987)

    Article  Google Scholar 

  11. Li, G.: A metric on space of measurable functions and the related convergence. Int. J. Uncertain. Fuzzy Knowl. Based Syst. 20, 211–222 (2012)

    MathSciNet  Article  Google Scholar 

  12. Hong, W.C.: Notes on Fréchet spaces. Int. J. Math. Math. Sci. 22, 659–665 (1999)

    MathSciNet  Article  Google Scholar 

  13. Ouyang, Y., Zhang, H.: On the space of measurable functions and its topology determined by the Choquet integral. Int. J. Approx. Reason. 52, 1355–1362 (2011)

    MathSciNet  Article  Google Scholar 

  14. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  15. Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)

    MATH  Google Scholar 

  16. Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)

    MathSciNet  Article  Google Scholar 

  17. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)

    MathSciNet  Article  Google Scholar 

  18. Siwiec, F.: On defining a space by a weak base. Pac. J. Math. 52, 233–245 (1974)

    MathSciNet  Article  Google Scholar 

  19. Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992)

    Article  Google Scholar 

  20. Wakker, P.P.: Prospect Theory for Risk and Ambiguity. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  21. Wang, Z.: The autocontinuity of set function and the fuzzy integral. J. Math. Anal. Appl. 99, 195–218 (1984)

    MathSciNet  Article  Google Scholar 

  22. Wu, C., Ren, X., Wu, C.: A note on the space of fuzzy measurable functions for a monotone space. Fuzzy Sets Syst. 182, 2–12 (2011)

    Article  Google Scholar 

  23. Zimper, A., Assa, H.: Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure. Math. Financ. Econ. 15, 353–380 (2021)

    MathSciNet  Article  Google Scholar 

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Correspondence to Alexander Zimper.

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Zimper, S., Zimper, A. On open \(\left( c,\epsilon \right) \)-balls in topological spaces that capture convergence in non-additive probability measure with probability-one coincidence. Afr. Mat. 33, 67 (2022).

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  • Convergence in non-additive probability measure
  • Weak-base topology
  • Dual-autocontinuity
  • Dual-nonatomicity

Mathematics Subject Classification

  • 60A86
  • 60B05