Abstract: Let \left(x_n, \mathscrF_n\right) be an integrable adapted sequence of \mathrmr . \mathrmv.'s, and \barT, T the sets of all stopping times and bounded stopping times, respectively. In this paper we prove that (1) if \left(x_n, \mathscrF_n\right) is a subpramart which satisfies the assumption \left(d_T^+\right): \frac\lim F E x_\tau^+< \infty or \left(O^+\right): \int_(\tau-\infty) x_\tau^+< \infty, \forall \tau \in \barT, then \left(x_n\right) converges a.e.; (2) if ( x_n, \mathscrF_n ) satisfies the assumption \left(C^+\right)or \left(d^+\right): \frac\lim n E x_n^+< \infty and \leftE\left(x_\tau \mid \mathscrF_n\right)-x_n\right^- \xrightarrowP 0 (pr. ), then ( x_n ) lower demiconverges a. \boldsymbol\theta..