I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows the result please let me know, many thanks!

Let $\{M_n\}_{n\geq 0}$ be a backward martingale, i.e.

$$E[M_n|\mathcal{F}_{n+1}]=M_{n+1},~ \forall n\geq 0$$

where $\mathcal{F}_n:=\sigma(M_k,~ k\geq n)$.

Now suppose

$$\lim_{n\to\infty}M_n=M,~ a.s.$$

My question is whether we have

$$\lim_{n\to\infty}E[|M_n-M|]=0$$

Clearly by Fatou Lemma and Jensen's inequality $M\in L^1$. If we suppose $M_0\in L^p$ with $p>1$ then it is just an application of Doob's inequality, but when $p=1$ I do not know how to prove it. Thanks a lot for your help!