= Integrable terminal bracket criterion
{title2=$\sup_t\mathbb E M_t^2<\infty\iff\mathbb E[M]_\infty<\infty$}
For a <continuous local martingale> starting at zero, finite expected terminal <quadratic variation> is equivalent to being an <L2-bounded continuous martingale>. Stop at bounded path and bracket levels and use $\mathbb E(M_t^\tau)^2=\mathbb E[M]_{t\wedge\tau}$. Uniform integrability removes the localization; <Doob L2 maximal inequality> justifies convergence of squares. The terminal identity is $\mathbb E M_\infty^2=\mathbb E[M]_\infty$. For nonzero initial values one must additionally assume $M_0\in L^2$.
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