The limit points of the strong law of large numbers under the sub-linear expectations

Abstract: Let ${X_n;n\ge 1}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean $\overline{\mu} $ and lower mean $\underline{\mu} $. Then for any Borel-measurable function $\varphi(x_1,\ldots,x_d)$ on $\mathbb R^d$ or continuous function $\varphi(x_1,x_2,\ldots)$ on $\mathbb R^{\mathbb N}$, $\sum_{i=1}^n X_i/n$ converges to $\underline{\mu}\wedge \varphi(X_1,X_2,\ldots)\wedge \overline{\mu}$ with upper capacity $1$. The limits of $\sum_{i=1}^nX_i/n$ can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of $(X_1, X_2,\ldots)$.