This is Problem 3.16 from Chapter 1 of Karatzas and Shreve.
Let $\{X_t, \mathscr{F}_t: 0 \le t < \infty\}$ be a right-continuous, nonnegative supermartingale; then $X_\infty(\omega) = \lim_{t \to \infty} X_t (\omega)$ exists for $P$-a.e. $\omega \in \Omega$, and $\{X_t , \mathscr{F}_t: 0 \le t \le \infty\}$ is a supermartingale.
I have shown that the limit exists a.e. from the same theorem for submartingales.
However, I am having difficulty showing that
$$E[X_\infty | \mathscr{F}_t] \le X_t.$$
I know this is the case if $\{X_t\}$ is uniformly integrable since then we have
$$
E[X_\infty | \mathscr{F}_t]
= \lim_{s \to \infty} E[X_s | \mathscr{F}_t]
\le X_t.
$$
So I want to show that $\int_{|X_t| > C} |X_t| dP \to 0$ as $C \to \infty$, but I got stuck here. How can we prove uniform integrability here?
Best Answer
Hint: There is no need to prove uniform integrability. Just apply Fatou's lemma (for conditional expectations).