Let (left(u_{n} ight)_{n in mathbb{N}}) be a sequence of positive integrable functions on a measure space ((X,

Let \(\left(u_{n}\right)_{n \in \mathbb{N}}\) be a sequence of positive integrable functions on a measure space \((X, \mathscr{A}, \mu)\). Assume that the sequence decreases to \(0: u_{1} \geqslant u_{2} \geqslant u_{3} \geqslant \cdots\) and \(u_{n} \downarrow 0\). Show that \(\sum_{n=1}^{\infty}(-1)^{n} u_{n}\) converges and is integrable, and that the integral is given by

\[\int \sum_{n=1}^{\infty}(-1)^{n} u_{n} d \mu=\sum_{n=1}^{\infty}(-1)^{n} \int u_{n} d \mu\]

[ mimic the proof of the Leibniz test for alternating series.]