Suppose I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$, adapted to a filtration $F_t$, that are martingales with respect to this filtration and that $\sup_n E[\sup_{0 \leq t \leq 1 } X_n(t)] < \infty$. Note that the filtration does not depend on $n$. I assume quite strong convergence of the $X_n(t)$ to a limit $X(t)$, namely that they converge uniformly on $[0,1]$ and for each $t$, $X_n(t)$ converges in $L^1$. I would like to know what are the weakest conditions I can assume about the $X_n(t)$ so that $X(t)$ will also be a martingale. If I assume the $X_n(t)$ are continuous or even left continuous then I believe that the limit will also be a martingale. If I just want to assume right continuity, I think I need that the fixed time distributions of $X(t)$ are nonsingular. What I'd really like to know is if some kind of continuity of the $X_n$'s is a requirement to push the martingale through to the limit.
