= L2-bounded martingale convergence theorem
{c}
{title2=$\sup_t\mathbb EM_t^2<\infty\Longrightarrow M_t\to M_\infty\text{ in }L^2$}
An $L^2$-bounded real <martingale> has a square-integrable terminal value, converges to it almost surely and in $L^2$, and satisfies $M_t=\mathbb E[M_\infty\mid\mathcal F_t]$. The $L^2$ Cauchy property follows from orthogonality of increments and monotonicity of $\mathbb EM_t^2$. This terminal representation permits optional sampling at finite unbounded <stopping times> and uniform-integrability control of stopped squares by conditional Jensen.
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