 Martingale convergence theorem

A discrete-time martingale with ${\sup }_n\mathbb{E}|M_n|<\infty$ converges almost surely to an integrable random variable. If the martingale is uniformly integrable, convergence also holds in $L^1$.

This is one of Doob's martingale convergence theorems; uniform integrability upgrades almost-sure convergence to convergence in mean.