= Solution
{c}
In any proposed decomposition, $Z_n'=X_n-Y_n'$ is integrable. Its increment is $\mathcal F_{n-1}$-measurable by predictability. Taking <conditional expectation> of the difference equation gives
$$
G_n=\mathbb E(X_n-X_{n-1}\mid\mathcal F_{n-1})
=\mathbb E(Y_n'-Y_{n-1}'\mid\mathcal F_{n-1})+(Z_n'-Z_{n-1}')
=Z_n'-Z_{n-1}'.
$$
The martingale increment has conditional mean zero, and a measurable integrable variable equals its own conditional expectation. Starting from $Z_0'=0$, summation gives $Z_n'=\sum_{j=0}^nG_j=Z_n$. Hence $Y_n'=X_n-Z_n'=Y_n$. Thus \b[the normalized predictable decomposition is unique], with a common probability-one event available for all integer times by countability.
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