In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a different filtration: $G_t = F_t \vee N$, where $N$ are the null sets of a probability space. The goal is to show $B(t)$ is a martingale w.r.t $G_{t+}$.
The hint given was to show that first it's a martingale with respect to $G_t$. I'm wondering if since $B(t)$ is a martingale w.r.t $F_t$ and $F_t \subset G_t $, is it automatic that $B(t)$ is a martingale w.r.t $G_t$ since if anything is measurable w.r.t $F_t$, it has to be measurable w.r.t $G_t$ (and by the same logic, it's a martingale with respect to $G_{t+}$? I feel like it requires more work than that but I'm not sure. Any hints would be helpful!