= Solution
Use conditional <monotone convergence> and the <tower property of conditional expectation> in the preceding construction:
$$
\begin{aligned}
\mathbb E[M_{n+1}\mid\mathcal F_n]
&=\lim_{p\to\infty}\mathbb E\bigl[\mathbb E[X_p^+\mid\mathcal F_{n+1}]\mid\mathcal F_n\bigr]\\
&=\lim_{p\to\infty}\mathbb E[X_p^+\mid\mathcal F_n]=M_n.
\end{aligned}
$$
This proves the <martingale> property. The construction gives $M_n\geq0$ and $\sup_n\mathbb EM_n\leq C$, so $M$ is bounded in $L^1$. Taking $p=n$ in the increasing family also gives $M_n\geq X_n^+\geq X_n$. Define $Y_n=M_n-X_n$. Then $Y$ is nonnegative and adapted, and
$$
\mathbb E[Y_{n+1}\mid\mathcal F_n]
=M_n-\mathbb E[X_{n+1}\mid\mathcal F_n]\leq M_n-X_n=Y_n.
$$
Thus $Y$ is a <supermartingale>, with $\mathbb E|Y_n|\leq\mathbb EM_n+\mathbb E|X_n|\leq2C$. The requested conclusion is
$$
\boxed{X_n=M_n-Y_n,\qquad M\text{ a nonnegative }L^1\text{-bounded martingale},\quad Y\text{ a nonnegative }L^1\text{-bounded supermartingale}.}
$$
This <positive martingale majorant of an L1-bounded submartingale> is not asserted to have <uniform integrability>; boundedness in $L^1$ alone does not imply that stronger property.
Back to article page
