Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 3 / e / Solution

Let $M=\mathbb{E}[Y\mid \mathcal{G}]$ and choose the stated strictly convex differentiable $\varphi$ with $|\varphi (x)|\le |x|$. Conditional Jensen inequality gives $\varphi (M)\le \mathbb{E}[\varphi (Y)\mid \mathcal{G}]$. Since $M\stackrel{d}{=}Y$, both sides have the same expectation, so equality holds almost surely. Strictness of the supporting-line inequality on $\{Y\ne M\}$ then forces $Y=M$ almost surely.