  In this paper we extend the definition of time conditional G-expectations
$\mathbb{\hat{E}}_{t}[\cdot]$ to a larger domain on which the dynamical
consistency still holds. In fact we can consistently define, by taking the
limit, the time conditional expectations for each random variable $X$ which is
the downward limit (resp. upward limit) of a monotone sequence $\{X_{i}\}$ in
$L_{G}^{1}(\Omega)$. To accomplish this procedure, some careful analysis is
needed. Moreover, we give a suitable definition of stopping times and obtain
the optional stopping theorem. We also provide some basic and interesting
properties for the extended conditional G-expectations.
