Let \(f: X ightarrow \mathbb{R}\) be a positive simple function of the form \(f(x)=\sum_{n=1}^{m} \xi_{n} \mathbb{1}_{A_{n}}(x), \xi_{n} \geqslant 0\), \(A_{n} \in \mathscr{A}\) - but not necessarily disjoint. Show that \(I_{\mu}(f)=\sum_{n=1}^{m} \xi_{n} \mu\left(A_{n}ight)\).

[use additivity and positive homogeneity of \(I_{\mu}\).]