Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 29 / 1 / a / Solution

Write $Y=\mathbb{E}[X\mid \mathcal{G}]$. The conditional expectation is an $L^2$ contraction by the conditional Jensen inequality, so $X,Y$ are square-integrable. The defining integral identity for conditional expectation, first with bounded measurable truncations of $Y$ and then with their $L^2$ limit, givesFor clarity, if $Y^{(K)}=\max (-K,\min (Y,K))$, then $\mathbb{E}[X{Y}^{(K)}]=\mathbb{E}[Y{Y}^{(K)}]$, and the Cauchy-Schwarz inequality permits the limit $K\to \infty$. Expanding the square now proves the equality case for conditional second moments:A nonnegative random variable of zero expected value vanishes almost surely. Therefore $\boxed{X=Y\text{ almost surely}.}$