Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $I\subseteq\mathbb R$ be countable, $(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ and $(M_t)_{t\in I}$ be a real-valued $L^1$-bounded uniformly integrable $(\mathcal F_t)_{t\in I}$-martingale on $(\Omega,\mathcal A,\operatorname P)$.
By the Dunford-Pettis theorem and the Eberlein-Šmulian theorem, $(M_t)_{t\in I}$ is relatively sequentially $\sigma\left((L^1(\operatorname P),L^\infty(\operatorname P)\right)$-compact and hence $$\left(M_{t_n}\right)_{n\in\mathbb N}\xrightarrow{\sigma\left((L^1(\operatorname P),L^\infty(\operatorname P)\right)}M_\infty\tag1$$ for some increasing $(t_n)_{n\in\mathbb N}\subseteq I$ and $M_\infty\in L^1(\mu)$; i.e. $$\forall Y\in L^\infty(\operatorname P):\operatorname E\left[M_{t_n}Y\right]\xrightarrow{n\to\infty}\operatorname E\left[M_\infty Y\right]\tag2.$$ From $(2)$ and the martingale property we immediately infer that $$\forall t\in I:\forall A\in\mathcal F_t:\forall n\in\mathbb N:t_n\ge t\Rightarrow\operatorname E\left[1_AM_t\right]=\operatorname E\left[1_AM_{t_n}\right]\tag3$$ and hence $$M_t=\operatorname E\left[M_\infty\mid\mathcal F_t\right]\;\;\;\text{for all }t\in I\tag4$$ by $(1)$.
Question 1: Let $\mathcal F_{\sup I}:=\sigma(\mathcal F_t:t\in I)$. How can we show that $M_\infty$ is $\mathcal F_{\sup I}$-measurable?
Question 2: If $(M_t)_{t\in I}$ is even $L^p(\operatorname P)$-bounded for some $p>1$, how can we show that $M_\infty\in L^p(\operatorname P)$?
Note that $\bigcup_{t\in I}\mathcal L^p(\mathcal F_t,\operatorname P)$ is a dense subspace of $\mathcal L^p(\mathcal F_{\sup I},\operatorname P)$ which should allow to conclude the desired claim in the first question.