Terminal expectation criterion for a nonnegative local martingale ($\mathbb{E}{Z}_{\infty }=\mathbb{E}{Z}_0⟹Z_t=\mathbb{E}[Z_{\infty }\mid {\mathcal{F}}_t]$)

A nonnegative local martingale is a supermartingale. If its integrable terminal limit has the same expectation as its initial value, conditional Fatou lemma gives $\mathbb{E}[Z_{\infty }\mid {\mathcal{F}}_t]\le Z_t$, and equality of expectations makes this an equality. It is therefore a uniformly integrable martingale.