Interchanging sums and integrals for absolutely summable series

Let \((X,\mcm,\mu)\) be a measure space and let \(f_n\colon X\to\CC\) be integrable for \(n\geq1\). The two quantities \[ \sum_{n=1}^\infty \int_X \abs{f_n}\dmu \quad\text{and}\quad \int_X\sum_{n=1}^\infty \abs{f_n}\dmu \] are equal, and if they are finite, then \(\sum_{n}f_n(x)\) converges absolutely for almost every \(x\), its sum is integrable, and \[ \int_X\sum_{n=1}^\infty f_n\dmu = \sum_{n=1}^\infty \int_X f_n\dmu . \]