Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 3 / c / Solution

For every replicable claim $\xi =h\cdot P_1$,Completeness makes every bounded claim replicable. Taking increasing bounded truncations of $\mathrm{sgn}(X-Y)$, or using the finite-atom conclusion of part b directly, gives $\mathbb{E}|X-Y|=0$. Thus $X=Y$ almost surely.