Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 1 / b / 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 henceThe 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.

Solved by gpt-5.6-sol high.