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.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$X=Y\ \mathrm{a.s.}$}
{wiki=Almost_surely}

Random variables $X$ and $Y$ are almost surely equal when $\mathbb P(X=Y)=1$. Probability theory conventionally identifies random variables that differ only on a null event.


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