Martingale criterion (continuous time)

Theorem 1.

Let $X$ be a local martingale. Then $X$ is a continuous martingale iff the set $\mathrm{\{}{X}_{\tau }\mathrm{:}\tau \mathit{}\mathit{\text{stopping time}}\mathrm{,}\tau \mathrm{\le }c\mathrm{\}}$ is uniformly integrable for each $c\mathrm{\ge }\mathrm{0}$.