= Solution
Define $A_0=0$ and recursively set
$$
A_{n+1}-A_n
=\mathbb E[X_{n+1}-X_n\mid\mathcal F_n].
$$
The increment is $\mathcal F_n$-measurable, so $A_{n+1}$ is $\mathcal F_n$-measurable by induction; hence $A$ is <predictable process>[predictable]. Conditional expectation preserves integrability, so $A_n$ and
$$
M_n=X_n-A_n
$$
are integrable and adapted. Their increments satisfy
$$
\mathbb E[M_{n+1}-M_n\mid\mathcal F_n]
=\mathbb E[X_{n+1}-X_n\mid\mathcal F_n]
-(A_{n+1}-A_n)=0,
$$
so $M$ is a martingale.
For uniqueness, suppose $X=M+A=M'+A'$ are two such decompositions. Then
$$
D=M-M'=A'-A
$$
is both a martingale and predictable, with $D_0=0$. Since $D_{n+1}$ is $\mathcal F_n$-measurable,
$$
D_{n+1}=\mathbb E[D_{n+1}\mid\mathcal F_n]=D_n.
$$
Induction gives $D=0$, and therefore $M=M'$ and $A=A'$. This proves the <Doob decomposition of an adapted integrable process>.
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