I just started learning about Brownian Motions and martin gales and have the following issue.
If $X_t$ is a brownian motion, I cannot understand how the results below are different when adapting for filtration $\mathcal{F}_s$.
$\mathbb{E[x^{2}_s | \mathcal{F}_s]} = x^{2}_s$ whereas $\mathbb{E[x^{2}_s]} = s$