Let $M_t$ be a continuous time real valued martingale, and $\mathcal F_t$ its natural filtration.

Suppose that $\mathcal F_t \setminus \mathcal F_s$ is nonempty for all $t > s$.

Let $\mathcal G$ be a sigma algebra, and define the filtration $\mathcal H_t := F_t \vee \mathcal G$.

**Question:** Is it true that $M$ is a $\mathcal H_t$ martingale if and only if $\mathcal G$ is independent of $\mathcal F_t$ for all $t$?

*Remark: The if direction follows from a monotone class argument.*