= Terminal decomposition of an L1-bounded supermartingale
{title2=$X_n=\mathbb E[X_\infty\mid\mathcal F_n]+Y_n,\quad Y_n\to0\text{ a.s.}$}
An $L^1$-bounded <supermartingale> has an <almost sure convergence> limit $X_\infty$ in $L^1$. The <conditional-expectation martingale> $M_n=\mathbb E[X_\infty\mid\mathcal F_n]$ has <uniform integrability> and converges to $X_\infty$ both <almost surely> and in $L^1$. To identify its limit, test against every event in every finite-time <sigma-algebra> and use uniqueness on their generated <sigma-algebra>. Then $Y=X-M$ is a <supermartingale> tending to zero <almost surely>. There is no general nonnegativity conclusion for $Y$, nor general <convergence in L1> to zero: the negative of the <fair-coin doubling martingale> is a counterexample to both stronger assertions.
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