as outlined in the title I wonder if we have
$M_1^n\rightharpoonup M_1$ implies $\mathbb{E}[M^n_1|\mathcal{F}_t]\rightharpoonup \mathbb{E}[M_1|\mathcal{F}_t]$ in $\mathbb{L}^2$.
I would say yes and do the proof as follows. Let $Y\in \mathbb{L}^2$ and define $M_t:=\mathbb{E}[M_1\mid\mathcal{F}_t]$ and $M_t^n:=\mathbb{E}[M_1^n\mid\mathcal{F}_t]$. We then get since $\mathbb{E}[Y\mid\mathcal{F}_t]\in\mathbb{L}^2$
$\mathbb{E}[YM^n_t]=\mathbb{E}[\mathbb{E}[M_1^n\mid\mathcal{F}_t]\mathbb{E}[Y\mid\mathcal{F}_t]]=\mathbb{E}[\mathbb{E}[M_1^n\mathbb{E}[Y\mid\mathcal{F}_t]\mid\mathcal{F}_t]]=\\ =\mathbb{E}[M_1^n\mathbb{E}[Y\mid\mathcal{F}_t]]\to\mathbb{E}[M_1\mathbb{E}[Y\mid\mathcal{F}_t]]=\mathbb{E}[M_tY] $,
where I did after the convergence the same steps as before.
I wonder if this is correct and the claim follows or if I overlooked anything?
Thank you in advance!
It is correct. Maybe the first step could be detailed a little bit more, since you do two thing at the same time: replace $M_t^n$ by its definition and use the definition of conditional expectation.