A similar question was asked here but I have a question about one of the steps.
To show that $M_t \in M := \{M_t : t \in [0,\infty)$ is integrable we consider $M_{t \wedge T_n}$ where $\{T_n\}$ are stopping times almost surey increasing to $\infty$. Then $M_{t \wedge T_n} \to M_t$ almost surely. We can now apply Fatou's Lemma to get that $$E(M_t) \le \liminf\limits _{n \to \infty}E(M_{t \wedge T_n}) = E(M_0)$$
But why does $\liminf\limits _{n \to \infty}E(M_{t \wedge T_n}) = E(M_0)$?
Best Answer
$M_{t\wedge T_n}: t \geq 0$ is a martingale for each $n$. So $EM_{t\wedge T_n}=EM_{0\wedge T_n}=EM_0$.