Let $X_n, n \geq 0$ be a $(\mathcal{F}_n)$-martingale.
We want to show that if $$X_n \overset{L^p}{\rightarrow} X_{\infty}$$ then there exists a $X \in L^p$ such that $$X_n = E[X \mid \mathcal{F}_n].$$
PROOF:
- As $X_n \rightarrow X_{\infty}$ in $L^p$, we have for $m \leq n$ that
$$X_m = E[X_n \mid \mathcal{F}_n]=:M_n.$$
- By Jensen's inequality for conditional expectation, we have
$$||M_n||_p = ||E[X_n \mid \mathcal{F}_n] ||_p \leq ||X_n||_p$$
- We have that $$E[X_n \mid \mathcal{F}_n] \overset{L^p}{\rightarrow} E[X_{\infty} \mid \mathcal{F}_n]$$ and therefore
$$X_m = E[X_{\infty}\mid \mathcal{F}_m].$$
QUESTIONS:
Actually I do neither understand the single steps nor the meaning of those. So could you please help me by adding some hints or some more Details? Also, I am not completely sure that there are no typos in it (concerning $m,n$). Thanks a lot!