Let $(\Omega,\mathcal{F},P)$ a probability space, $\{\mathcal{F}_t\}_{t\in N_0}$ a discrete filtration and $\{Q_t\}$ a sequence of probabilities equivalent to $P$ and such that $Q_t|_{\mathcal{F}_{t-1}}=Q_{t-1}|_{\mathcal{F}_{t-1}}$ for each $t\in N$.
Let $\mathcal{F}_\infty:=\sigma(\underset{t\in N_0}\cup\mathcal F_t)$.
Then, for each $A\in \underset{t\in N_0}\cup\mathcal F_t$, we define $Q(A):=\underset{t\rightarrow\infty}\lim Q_t(A)$.
So, clearly $Q|_{\mathcal{F}_t}=Q_t|_{\mathcal{F}_t}$.
The question is: under which conditions can $Q$ be extended to the $\sigma$-algebra $\mathcal{F}_\infty$ and define a probability measure?
The union $$\bigcup_{t\in N_0}\mathcal F_t$$ is an algebra (although not necessarily a $\sigma$-algebra), so you can use Carathéodory’s theorem to establish the existence of a unique extension.
The only thing left to check is whether $Q$ is countably additive on $$\bigcup_{t\in N_0}\mathcal F_t,$$ that is, whether $$\lim_{t\to\infty}\sum_{k\in\mathbb N}Q_t(A_k)=\sum_{k\in\mathbb N}\lim_{t\to\infty}Q_t(A_k)$$ whenever $$A_k\in\bigcup_{t\in N_0}\mathcal F_t\quad\text{for each $k\in\mathbb N$}$$ and $$\bigcup_{k\in\mathbb N}A_k\in\bigcup_{t\in N_0}\mathcal F_t,$$ and the union is disjoint.