= Positive martingale majorant of an L1-bounded submartingale
{title2=$M_n=\lim_{p\to\infty}\mathbb E[X_p^+\mid\mathcal F_n]$}
For a <submartingale> with $\sup_n\mathbb E|X_n|=C<\infty$, its <positive part> is a <submartingale> by the <conditional Jensen inequality>. Thus the displayed <conditional expectations> increase in $p$. Their limits are finite <almost surely> and have <expectations> at most $C$, by the <monotone convergence theorem>. Conditional monotone convergence and the <tower property of conditional expectation> give $\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n$. Moreover $M_n\geq X_n^+$. Consequently $Y_n=M_n-X_n$ is a nonnegative <supermartingale>, $\sup_n\mathbb E Y_n\leq2C$, and $X=M-Y$. The resulting <martingale> need not have <uniform integrability>.
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