Conditional Jensen inequality

Let $X$ be a real integrable random variable, let $\mathcal{G}$ be a sub-sigma-algebra, and let $\varphi :\mathbb{R}\to \mathbb{R}$ be a finite convex function with $\varphi (X)$ integrable. Thenalmost surely. These explicit integrability assumptions make both conditional expectations well defined.

A finite convex function on the real line is the supremum of a countable family of supporting affine functions, for example choosing supports at rational points and using continuity. Write them as $a_{j}x+b_j$. For each $j$, monotonicity and linearity of conditional expectation give $a_j\mathbb{E}[X\mid \mathcal{G}]+b_j\le \mathbb{E}[\varphi (X)\mid \mathcal{G}]$. Countability permits one exceptional null set for every $j$. Taking the supremum proves the inequality.

With $\varphi (x)=x^2$ and square-integrable $X$, the inequality says that retaining only the information in $\mathcal{G}$ cannot increase its second moment. More generally it is the convex-loss step behind the Rao-Blackwell theorem.