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.