  Using the spectral measure $\mu_\mathbb{S}$ of the stopping time
$\mathbb{S},$ we define the stopping element $X_\mathbb{S}$ as a Daniell
integral $\int X_t\,d\mu_\mathbb{S}$ for an adapted stochastic process
$(X_t)_{t\in J}$ that is a Daniell summable vector-valued function. This is an
extension of the definition previously given for right-order-continuous
sub-martingales with the Doob-Meyer decomposition property. The more general
definition of $X_\mathbb{S}$ necessitates a new proof of Doob's optional
sampling theorem, because the definition given earlier for sub-martingales
implicitly used Doob's theorem applied to martingales. We provide such a proof,
thus removing the heretofore necessary assumption of the Doob-Meyer
decomposition property in the result.
  Another advancement presented in this paper is our use of unbounded order
convergence, which properly characterizes the notion of almost everywhere
convergence found in the classical theory. Using order projections in place of
the traditional indicator functions, we also generalize the notion of uniformly
integrable sequences. In an essential ingredient to our main theorem mentioned
above, we prove that uniformly integrable sequences that converge with respect
to unbounded order convergence also converge to the same element in
$\mathcal{L}^1$.
