I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$. Suppose that there is a filtration $F_t$ such that for each $n$, $X_n$ is a martingale with respect to this filtration. Note that the filtration does not depend on $n$. Also assume that $$\sup_n E[\sup_{0 \leq t \leq 1 } |X_n(t)|] < \infty.$$

Finally suppose for every $t \in [0,1]$ $X_n(t)$ converge uniformly to a limit $X(t)$ and $X_n(t)$ 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 non-singular. 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.