Solution
= Solution
Uniqueness means <almost sure equality>. Suppose that two $\mathcal F_n$-measurable integrable random variables $Y$ and $Z$ satisfy the integral identity from part (a). The event $D=\{Y>Z\}$ belongs to $\mathcal F_n$, and hence
$$
\mathbb E[(Y-Z)\mathbf1_D]=0.
$$
The integrand is nonnegative and is positive precisely on $D$, so $\mathbb P(D)=0$. Interchanging $Y$ and $Z$ gives $\mathbb P(Z>Y)=0$, and therefore $Y=Z$ almost surely.