## Abstract

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.

This is a preview of subscription content, access via your institution.

## Notes

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.

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 \(\lambda \).

## References

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

Arhangel’skiĭ, A.V.: Mappings and spaces. Russ. Math. Surv.

**21**, 115–162 (1966)Assa, H., Zimper, A.: Preferences over all random variables: incompatibility of convexity and continuity. J. Math. Econ.

**75**, 71–83 (2018)Billingsley, P.: Probability and measure. Wiley, New York (1995)

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)Choquet, G.: Theory of capacities. Annales de l’Institut Fourier

**5**, 131–295 (1954)Day, M.M.: The Spaces \(L^{p}\) with \(0<p<1\). Bull. Am. Math. Soc.

**46**, 816–823 (1940)Denneberg, D.: Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht (1994)

Ellsberg, D.: Risk, ambiguity and the savage axioms. Quart. J. Econ.

**75**, 643–669 (1961)Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ.

**16**, 65–88 (1987)Li, G.: A metric on space of measurable functions and the related convergence. Int. J. Uncertain. Fuzzy Knowl. Based Syst.

**20**, 211–222 (2012)Hong, W.C.: Notes on Fréchet spaces. Int. J. Math. Math. Sci.

**22**, 659–665 (1999)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)Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

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

Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc.

**97**, 255–261 (1986)Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica

**57**, 571–587 (1989)Siwiec, F.: On defining a space by a weak base. Pac. J. Math.

**52**, 233–245 (1974)Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain.

**5**, 297–323 (1992)Wakker, P.P.: Prospect Theory for Risk and Ambiguity. Cambridge University Press, Cambridge (2010)

Wang, Z.: The autocontinuity of set function and the fuzzy integral. J. Math. Anal. Appl.

**99**, 195–218 (1984)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)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)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are very grateful to an anonymous referee for helpful comments and suggestions.

## Rights and permissions

## About this article

### Cite this article

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). https://doi.org/10.1007/s13370-022-01001-z

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/s13370-022-01001-z

### Keywords

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

### Mathematics Subject Classification

- 60A86
- 60B05