Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, finitely-additive set function on $\mathcal{F}$ is also countably-additive, thus can be extended to a probability measure on $\mathcal{B}$ uniquely by Caratheodory's extension theorem.
Let $m_n$ be a sequence of probability measures on $\mathcal{B}$. If for all $F\in\mathcal{F}$, $m_n(F)\to\alpha(F)$ for some set function $\alpha$, then $\alpha$ is normalized, non-negative, finitely-additive on $\mathcal{F}$ and so can be extended to a probability measure $m$ on $\mathcal{B}$. Is it true that $m_n(F)\to m(F)$ for all $F\in\mathcal{B}$?
This is related to the following question: if $\mathcal{F}\subset\mathcal{G}\subset\mathcal{B}$ where $\mathcal{G}$ could be countable, and the convergence $m_n(F)\to\alpha(F)$ is on $\mathcal{G}$. Restricting $\alpha$ on $\mathcal{F}$ and extend $\alpha|_{\mathcal{F}}$ to a probability measure $m$ on $\mathcal{B}$. Then is it true that $m(F) = \alpha(F)$ on $\mathcal{G}$? We know it is true if $\mathcal{G}=\mathcal{B}$, since by Vitali–Hahn–Saks theorem, $\alpha$ is already a probability measure.
