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}

Question:

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 $I_{\mu}$.]

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