Given the mentioned stochastic processes $M_n$, $X_n$ I tried to show that $M_n$ is a martingale with respect to the filtration $\mathcal{F_n}$ generated by $M_n$. My proof is dependent on the fact that: $$E[X_k|\mathcal{F_n}] = E[X_k]$$ i.e $X_k$ is independent of $\mathcal{F_n}$.
My question is: is this true, and if so, what is the reasoning?