= Solution
Relative to a <filtration> $(\mathcal F_t)$, a <martingale> is an adapted integrable process satisfying $\mathbb E[M_t\mid\mathcal F_s]=M_s$ for $s\le t$. The <martingale convergence theorem> gives an almost-sure finite limit for a nonnegative martingale, with $\mathbb E[M_\infty]\le\mathbb E[M_0]$ by <Fatou's lemma>. In continuous time take the usual right-continuous version and filtration hypotheses. Convergence in $L^1$, and preservation of expectation at infinity, additionally require <uniform integrability>.
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