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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

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.

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Notes

  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].

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

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  • 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