= Solution
The <almost sure submartingale convergence theorem> says that if a <submartingale> $(Y_n)$ satisfies
$$
\sup_n\mathbb E[Y_n^+]<\infty,
$$
then there is an <integrable random variable> $Y_\infty$ such that
$$
\boxed{Y_n\longrightarrow Y_\infty\quad\text{almost surely}.}
$$
Indeed the positive-part bound also bounds the negative parts, since $\mathbb E Y_n\geq\mathbb E Y_0$; the theorem gives a finite limit. In particular, the <martingale convergence theorem> applies to any <martingale> with $\sup_n\mathbb E|M_n|<\infty$.
For a nonnegative <martingale>, $\mathbb E M_n=\mathbb E M_0$ supplies this bound automatically. Its limit obeys $\mathbb E M_\infty\leq\mathbb E M_0$ by the <Fatou lemma>. \b[<Almost sure convergence> alone need not preserve the <expectation> or give convergence in L1.] <Uniform integrability> is the additional condition that supplies <convergence in L1> and the identity $M_n=\mathbb E[M_\infty\mid\mathcal F_n]$.
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