$\sigma(\xi)$ is independent of a fixed $\sigma$-algebra

Question

I want to prove that if $\lim_{n->\infty}\xi_n = \xi$ pointwise and each $\xi_n$ is independent from a fixed sigma-algebra $F$ then $\xi$ is independent from $F$.

I understand that $\sigma(\xi)$ will be independent from $F$ too.

How can it be proved: if $\sigma(\xi)$ is independent from $F$ then $\xi$ is independent of $F$?

Answer 1

$\sigma (\xi)=\{\xi^{-1} (A): A \in \mathcal B\}$ where $\mathcal B$ is the Borel sigma algebra of $\mathbb R$. The hypothesis gives $P(\xi ^{-1}(A) \cap E)=P(\xi ^{-1}(A) )P(E)$ for any Borel set $A$ and any $E \in F$. This is the definition of $\xi$ being independent of $F$.