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Define $A_0=0$ and recursively setThe increment is ${\mathcal{F}}_n$-measurable, so $A_{n+1}$ is ${\mathcal{F}}_n$-measurable by induction; hence $A$ is predictable. Conditional expectation preserves integrability, so $A_n$ andare integrable and adapted. Their increments satisfyso $M$ is a martingale.

For uniqueness, suppose $X=M+A=M^{\prime }+A^{\prime }$ are two such decompositions. Thenis both a martingale and predictable, with $D_0=0$. Since $D_{n+1}$ is ${\mathcal{F}}_n$-measurable,Induction gives $D=0$, and therefore $M=M^{\prime }$ and $A=A^{\prime }$. This proves the Doob decomposition of an adapted integrable process.