Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and \(\mathscr{G} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra. Show that, in general,

\[\int \mathbb{E}^{\mathscr{G}} u d \mu \leqslant \int u d \mu, \quad u \in L^{1}(\mathscr{A}), u \geqslant 0\]

with equality holding only if \(\left.\muight|_{\mathscr{G}}\) is \(\sigma\)-finite.