Let \((X, \mathscr{A}, \mu)\) be a measure space, \(\mathscr{G} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra and let \(u:=f \mu\) where \(f \in \mathcal{M}^{+}(\mathscr{A})\) is a density \(f>0\). Denote by \(\mathbb{E}_{u}^{\mathscr{G}}\), resp. \(\mathbb{E}_{\mu}^{\mathscr{G}}\), the projections in the spaces \(L^{2}(\mathscr{A}, \mu)\), resp. \(L^{2}(\mathscr{A}, u)\).

(i) Let \(G^{*}:=\left\{\mathbb{E}_{\mu}^{\mathscr{G}} f>0ight\}\) and show that \(\left.uight|_{G^{*} \cap \mathscr{G}}=\left.\left(\mathbb{E}_{\mu}^{\mathscr{G}} fight) \muight|_{G^{*} \cap \mathscr{G}}\left(G^{*} \cap \mathscr{G}ight.\) denotes the trace \(\sigma\)-algebra).

(ii) Show that \(P u:=\mathbb{1}_{G^{*}} \mathbb{E}_{\mu}^{\mathscr{G}}(f u) / \mathbb{E}_{\mu}^{\mathscr{G}} f\) maps bounded \(L^{2}(\mathscr{A}, u)\)-functions into \(L^{2}(\mathscr{G}, u)\) and satisfies \(\|P u\|_{L^{2}(\mathscr{G}, u)} \leqslant\|u\|_{L^{2}(\mathscr{A}, \mu)}\).

(iii) Show that \(P=\mathbb{E}_{u}^{\mathscr{G}}\)

(iv) When do we have \(\mathbb{E}_{\mu}^{\mathscr{G}} u=\mathbb{E}_{u}^{\mathscr{G}} u\) for all \(u \in L^{2}(\mathscr{A}, \mu) \cap L^{2}(\mathscr{A}, u)\) ?

Remark. The above result allows us to study conditional expectations for finite measures \(\mu\) only and to define for more measures a conditional expectation by

\[\mathbb{E}_{u}^{\mathscr{G}} u:=\frac{\mathbb{E}_{\mu}^{\mathscr{G}}(f u)}{\mathbb{E}_{\mu}^{\mathscr{G}} f} \mathbb{1}_{\left\{\mathbb{E}_{\mu}^{\mathscr{G}} f>0ight\}}\]