Suppose I have a sequence $\{X_t^n,{\cal F}_t^n\}$ of martingales (with respect to different filtrations), and suppose that $X_n$ converge weakly to a continuous process $X$. What are the conditions for $X$ to be a martingale with respect to its natural filtration ${\cal F}_t=\sigma(X_s,0\leq s\leq t)$?
In the problem I have in my research I can show that the $\{X^n\}$ are uniform integrable. So, in order to show $E[C_{t+u}1_A]=E[C_t1_A]$ for all $A\in{\cal F}_t$, I can show first $$E[C_{t+u}g(C_s,0\leq s\leq t)]=E[C_{t}g(C_s,0\leq s\leq t)]$$ for all continuous and bounded $g$.
Then, I'm using the functional monotone class theorem (link); but I'm missing one step, which is to show that the $\sigma$-field generated by all bounded and continuous functions of $\{C_s,0\leq s\leq t\}$ is ${\cal F}_t$. I'm pretty sure this is true but I don't know how to prove this. I think this will answer my question.
I hope this question makes sense...
Thanks
In the book: "J. Jacod and A. N. Shiryayev, Limit Theory for Stochastic Processes, 2ed. Springer, 2003'', There is following result(p.522 Propsition 1.1): Assume that $ (M^n) $ is a sequence of martingales converging in law to a limit process $ M $, and that $ |M^n|\le b $ identically for some constant $ b $. Then $ M $ is a martingale with respect to the filtration that it generates.