Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 1 / e

If $\mathcal{G}\subseteq \mathcal{H}$, then $\mathbb{E}[X\mid \mathcal{G}]$ is already $\mathcal{H}$-measurable, soIf $\mathcal{H}\subseteq \mathcal{G}$, the tower property of conditional expectation givesFinally, if $\mathcal{G}$ and $\mathcal{H}$ are independent, the $\mathcal{G}$-measurable variable $\mathbb{E}[X\mid \mathcal{G}]$ is independent of $\mathcal{H}$. Its conditional expectation given $\mathcal{H}$ is its mean $\mathbb{E}[X]$. Part c shows that the right side is also $\mathbb{E}[X]$.