  We give a very simple and elementary proof of the existence of a weakly
compact family of probability measures $\{P_{\theta}:\theta \in \Theta \}$ to
represent an important sublinear expectation--G-expectation
$\mathbb{E}[\cdot]$. We also give a concrete approximation of a bounded
continuous function $X(\omega)$ by an increasing sequence of cylinder functions
$L_{ip}(\Omega)$ in order to prove that $C_{b}(\Omega)$ belongs to the
$\mathbb{E}[|\cdot|]$-completion of the $L_{ip}(\Omega)$.

Probability