I would like to understand which relationships hold among the sigma algebras $\sigma(X, X^2)$, $\sigma(X)$ and $\sigma(X^2)$, where X is a random variable. I would expect that $\sigma(X, X^2)=\sigma(X)$. If this is true, then is $X^2$ $\sigma(X)$-measurable?
Moreover, I would like to know if there exist more sigma algebras w.r.t. a process is adapted. I am thinking that if $X_i$ are i.i.d. than $S_n=\sum_{i=1}^n X_i$ is $\mathcal{F}_n:=\sigma(S_n)$-measurable but also $\mathcal{G}_n:=\sigma(X_1, X_2, \dots, X_n)$-measurable. What changes if I take $\{\mathcal{F}_n\}$ instead of $\{\mathcal{G}_n\}$ as a filtration for the process $\{S_n\}$? Because for both the process is a martingale!