Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Assume that $\mathcal F_k$ is generated by a countable partition for every $k\in\mathbb N.$ Let $(\mathbb Q_k)_{k\in\mathbb N}$ be a sequence of probability measures which is weak$^*$-convergent to a probability measure $\mathbb Q.$
The weak$^*$-topology is the weakest topology such that all the linear functionals $L_Z:\mathbb P\mapsto \int_\Omega Zd\mathbb P$ for $Z:Ω\rightarrow\mathbb R$ $\mathcal F$-measurable and bounded, are continuous.
Assume that for all $k\in\mathbb N$ we have $$E_{\mathbb Q_k}[X\mid\mathcal F_k]=0.$$ Can we conclude that $$\ E_{\mathbb Q'}[X\mid\mathcal F]=0$$ for some probability measure $\mathbb Q'$ which is equivalent to $\mathbb Q$? (maybe this works even for $\mathbb Q$ itself). Note that by Doob's martingale convergence theorem, we have $$\lim_{l\rightarrow\infty}E_{\mathbb Q_k}[X\mid\mathcal F_l]= E_{\mathbb Q_k}[X\mid\mathcal F]$$ in $L^1$ and $\mathbb Q_k$-a.s., for each $k\in\mathbb N.$ I am also not sure if weak$^*$-convergence implies that $$\lim_{k\rightarrow\infty} E_{\mathbb Q_k}[X\mid\mathcal F]=E_{\mathbb Q}[X\mid\mathcal F]$$ $\mathbb Q$-a.s. or in $L^1(\mathbb Q)?$