Random variables and independence of $\sigma$-algebras If a random variable $X$ is independent from the $\sigma$-algebra $F_t$ for every $t$ in a collection of indexes, is it true that $X$ in independent from the $\sigma$-algebra generated by all the $F_t$?
No.  For example, you can have three random variables $X, Y, Z$ such that 
$X$ is independent of $Y$ and independent of $Z$ but completely determined by $Y$ and $Z$.  The standard example involves two coin-tosses.
Afaik, if we have the additional assumption that $\mathscr F_{t_1}$ and $\mathscr F_{t_2}$ are independent for all distinct indices $t_1$ and $t_2$, we don't have the independence of $X$ and $\sigma(\bigcup_{t \in I} \mathscr F_t)$.