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

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$ with the finite Choquet expectation, upper mean $\bar{\mu }$ and lower mean $\underset{‾}{\mu }$. Then for any Borel-measurable function $\phi (x_1,\dots ,x_d)$ on ${\mathbb{R}}^d$ or continuous function $\phi (x_1,x_2,\dots )$ on ${\mathbb{R}}^{\mathbb{N}}$, ${\sum }_{i=1}^n{X}_i\mathrm{/}n$ converges to $\underset{‾}{\mu }\wedge \phi (X_1,X_2,\dots )\wedge \bar{\mu }$ with upper capacity $1$. The limits of ${\sum }_{i=1}^n{X}_i\mathrm{/}n$ can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of $(X_1,X_2,\dots )$.