= Terminal expectation criterion for a nonnegative local martingale
{title2=$\mathbb EZ_\infty=\mathbb EZ_0\;\Longrightarrow\;Z_t=\mathbb E[Z_\infty\mid\mathcal F_t]$}
A <nonnegative local martingale> is a <supermartingale>. If its integrable terminal limit has the same expectation as its initial value, conditional <Fatou lemma> gives $\mathbb E[Z_\infty\mid\mathcal F_t]\leq Z_t$, and equality of expectations makes this an equality. It is therefore a <uniformly integrable martingale>.
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