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

Stopped processes and Doob's optional sampling theorem