= Almost sure submartingale convergence theorem
{title2=$\sup_n\mathbb E Y_n^+<\infty\Longrightarrow Y_n\to Y_\infty\text{ a.s.}$}
= Submartingale almost sure convergence theorem
{synonym}
A <submartingale> with $\sup_n\mathbb E Y_n^+<\infty$ converges almost surely to a finite <integrable random variable>. The positive-part bound also bounds negative parts because $\mathbb E Y_n\geq\mathbb E Y_0$. The result specializes to an $L^1$-bounded <martingale>, and hence to every nonnegative <martingale>. It does not by itself give <convergence in L1>; <uniform integrability> supplies that stronger conclusion.
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