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

Conditional expectation is the orthogonal projection from $L^2(\mathcal{F})$ onto $L^2(\mathcal{G})$. The difference $X-\mathbb{E}[X\mid \mathcal{G}]$ is orthogonal to every $\mathcal{G}$-measurable square-integrable variable, including $\mathbb{E}[X\mid \mathcal{G}]-\mathbb{E}[X\mid \mathcal{H}]$. The Pythagorean theorem in an inner-product space applied togives the identity.