Given a sequence of $\mathbb{R}-$valued random variables $\{X_n\}_{n\in\mathbb N}\in L^1$ such that $X_n\to X$ in $L^1$ for some $X\in L^1$. Let $Y\in L^1$, then it is well-known that $E[X_n|Y]\to E[X|Y]$ in $L^1$. But I'm wondering whether we have $$E[Y|X_n]\to E[Y|X]\ \ \text{in}\ L^1?$$
I've tried some examples such as Bernoulli random variables and it seems that it is right in some sense, but I really cannot figure a proof. I don't know how to handle the case where the conditional $\sigma-$fields change.
Any help would be appreciated.
Let $Y$ be a non-constant r.v. in $L^{1}$ and $X_n=\frac Y n$. Then $X_n \to X$ in $L^{1}$ where $X=0$. Now $E[Y|X_n]=Y$ for all $n$ and $E[Y|X]=EY$.