I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$.
We know there exists $X_{\infty}$ such that $X_n\to X_{\infty}$ a.s.
I want to show that for $Y\in \mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$ we have $X_n \overset{\mathcal{L}^2}{\to} X_{\infty}$. I hope the following line of reasning is correct.
\begin{align*}\int_{\Omega}|X_n-X_{\infty}|^2d\mathbb{P} &= \int_{\Omega}|X_n^2+X_{\infty}^2-2X_nX_{\infty}|d\mathbb{P}\\&\leq \int_{\Omega}|X_n||X_n-X_{\infty}|d\mathbb{P}+\int_{\Omega}|X_{\infty}||X_{\infty}-X_n|d\mathbb{P}\end{align*} which converges to 0 by $\mathcal{L}^1$-convergence.
Then I want to find a condition s.t. $X_{\infty}= Y$, but isn't this simply saying that $\lim_n \mathcal{F}_n = \mathcal{F}$. ?
Also, i want to find an example where $\mathbb{P}(X_{\infty}= Y)=0$. So, this means $\lim_n \mathbb{P}(\mathbb{E}[Y|\mathcal{F}_n]= Y)=0$. Hence $\mathbb{P}\left\{\omega:\mathbb{E}[Y|\mathcal{F}_n](\omega)= Y(\omega)\right\}=0 $ for all n.
How do I find such $Y$ and $\mathcal{F}_n$?
Thank you kindly in advance