 Conditional Jensen inequality

For an integrable random variable $X$, a sub-sigma-algebra $\mathcal{G}$, and a convex function $\varphi$ with suitable integrability, $\varphi (\mathbb{E}[X\mid \mathcal{G}])\le \mathbb{E}[\varphi (X)\mid \mathcal{G}]$ almost surely. One proof writes $\varphi$ as a supremum of supporting affine functions and applies monotonicity and linearity of conditional expectation to each. This proves the convex-loss form of the Rao-Blackwell theorem.