Let \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be a sequence of disjoint sets such that \(\bigcup_{n \in \mathbb{N}} A_{n}=X\). Show that for every \(u \in \mathcal{M}^{+}(\mathscr{A})\)

\[
\int u d \mu=\sum_{n=1}^{\infty} \int \mathbb{1}_{A_{n}} u d \mu
\]

Use this to construct on a \(\sigma\)-finite measure space \((X, \mathscr{A}, \mu)\) a function \(w\) which satisfies \(w(x)>0\) for all \(x \in X\) and \(\int w d \mu<\infty\).