Stopped-martingale uniform integrability criterion
= Stopped-martingale uniform integrability criterion
For an <almost surely> finite <stopping time> $T$, if $\mathbb E|M_T|<\infty$ and $\mathbb E[|M_n|\mathbf1_{\{T>n\}}]\to0$, then $M_{n\wedge T}\to M_T$ with <convergence in L1>. Thus <L1 convergence implies uniform integrability> makes the <stopped martingale> <uniformly integrable>.