Let $\{X_n\}$ be a sequence of random variables. $X_n\rightarrow X$ a.s. Let $G$ be a $\sigma$-algebra. Suppose each $X_n$ is independent of $G$. Is it true that $X$ is also independent of $G$?
I proved it in this way:
Fact 1 a random variable $X$ is independent of a $\sigma$-algebra $G$ if and only if $\forall A\in G$, $X$ is independent of $1_A$.
Fact 2 Kac's theorem: two random variables $X,Y$ are independent if and only if $\forall \eta,\xi \in \mathbb{R}^d: \mathbb{E}e^{i (X,Y) \cdot (\xi,\eta)} = \mathbb{E}e^{i X \cdot \xi} \cdot \mathbb{E}e^{i Y \cdot \eta}$
By the two facts above, we can let $X=X_n$ and $Y=1_A$ and use dominant convergence theorem to finish the proof.
So my question is:
Can somebody help verify the proof or provide some suggestions?
Is there any alternative approach instead of using Kac's theorem(Which I failed to find this theorem on Google)? Thank you very much.
