= Uniformly integrable martingale
{title2=$M_t=\mathbb E[M_\infty\mid\mathcal F_t]$}
A <martingale> whose values over its whole time interval form a <uniformly integrable> family has an integrable terminal limit and is closed by that limit. Conversely, conditional expectations of an integrable terminal variable form a <uniformly integrable martingale>, by <uniform integrability of conditional expectations>. For <continuous martingales>, one may equivalently use <uniform integrability> of the family stopped at finite stopping times.
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