Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\mathscr{H} \subset \mathscr{G}\) be two sub- \(\sigma\)-algebras. Show that

\[\mathbb{E}^{\mathscr{H}} \mathbb{E}^{\mathscr{G}} u=\mathbb{E}^{\mathscr{G}} \mathbb{E}^{\mathscr{H}} u=\mathbb{E}^{\mathscr{H}} u \quad \forall u \in L^{2}(\mathscr{A}) .\]