Fix a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\geq 0},\mathbb{P})$ and an $L^1$-bounded submartingale $X_n$.
We can show that, for $n\geq 0$, the sequence $(\mathbb E[X^+_p|\mathcal{F}_n],p\geq n)$ is increasing and converges to an a.s. limit $M_n$. We want to show that $(M_n)_{n\geq 0}$ is an $L^1$-bounded martingale.
$\mathbb E[M_{n+1}|\mathcal{F}_n]=\mathbb E\lim_{p\to\infty}[\mathbb E[X_p^+|\mathcal{F}_{n+1}]|\mathcal{F}_n]$
So it seems the right tool here is dominated convergence. By the conditional dominated convergence theorem, $M$ is a martingale if, for $n\geq0$, $\mathbb E|M_n|<\infty$.
How do we show that $M$ is $L^1$-bounded?
Thank you.