If $X$ and $Y$ are both independent from $\mathscr {F}$, is $f(X,Y)$ also independent from $\mathscr{F}$?
$X_1,...,X_n$ be independent RVs and $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$. Show that $\sigma (X_1,...,X_n) \perp \mathcal F$