Let $M_t$ be an $\mathcal{F}_t$ martingale such that $\sup_t E|M_t|^p < \infty$ for some $p>1$. Show there exists a $Y\in L^1(P)$ such that $M_t = E[Y|\mathcal{F}_t]$.
I am not sure how to answer this question.
The problem suggests using the result that: the condition in the problem implies the existence of a limiting random variable $M \in L^1(P)$ such that $M_t \to M$ almost surely and $$\int |M_t - M|dP \to 0$$
as $t \to \infty$.
But I'm not sure how to represent $M_t = E[g(M)|\mathcal{F}_t]$ given the corollary doesn't tell us the relationship between $M_t$ and $M$ other than that they converge almost everywhere.
Since $(M_t)$ is a uniformly integrable martingale, it converges a.s. and in $L^1$ to some limit $M$. Then for any $F\in \mathcal{F}_t$ and $s>t$, $$ \mathsf{E}[M_t1_F]=\mathsf{E}[M_s1_F]\to\mathsf{E}[M1_F]. $$ Therefore, $M_t=\mathsf{E}[M\mid \mathcal{F}_t]$ a.s.