  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 $(X_1, X_2,\ldots)$.
