Let \((X, \mathscr{A}, \mu)\) be a measure space and \(1 \leqslant p<\infty\). Show that \(f \in \mathcal{E}(\mathscr{A}) \cap \mathcal{L}^{p}(\mu)\) if, and only if, \(f \in \mathcal{E}(\mathscr{A})\) and \(\mu\{f eq 0\}<\infty\).

In particular, \(\mathcal{E}(\mathscr{A}) \cap \mathcal{L}^{p}(\mu)=\mathcal{E}(\mathscr{A}) \cap \mathcal{L}^{1}(\mu)\).