= Solution
A <filtration> $(\mathcal F_n)_{n\geq0}$ on a <probability space> is an increasing sequence of <sigma-algebras> contained in $\mathcal F$: $\mathcal F_n\subseteq\mathcal F_{n+1}$. A real <stochastic process> $(X_n)$ is a <martingale> for this <filtration> if $X_n$ is $\mathcal F_n$-measurable, $\mathbb E|X_n|<\infty$, and
$$
\boxed{\mathbb E[X_{n+1}\mid\mathcal F_n]=X_n\quad\text{almost surely for every }n.}
$$
The first condition is <adaptedness>; the <tower property of conditional expectation> then gives $\mathbb E[X_n\mid\mathcal F_m]=X_m$ whenever $m\leq n$.
The almost-sure <martingale convergence theorem> states that if a real <martingale> satisfies $\sup_n\mathbb E|X_n|<\infty$, then some integrable <random variable> $X_\infty$ satisfies $X_n\to X_\infty$ <almost surely>. The more general <almost sure submartingale convergence theorem> needs only $\sup_n\mathbb E[X_n^+]<\infty$ for a <submartingale>. In particular, a nonnegative <martingale> converges to a finite integrable limit <almost surely>, because its <expected value> is constant. Mere boundedness in the <L1 norm> does not imply <convergence in L1>. The stronger <uniformly integrable martingale convergence theorem> gives both almost-sure convergence and <convergence in L1> when the <martingale> is <uniformly integrable>.
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