Solution
= Solution
Let $M=\mathbb E[Y\mid\mathcal G]$ and choose the stated strictly convex differentiable $\phi$ with $|\phi(x)|\leq|x|$. Conditional <Jensen inequality> gives $\phi(M)\leq\mathbb E[\phi(Y)\mid\mathcal G]$. 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.