Absolutely summable series in $L^1$ converge almost everywhere and in $L^1$

Let \((X,\mcm,\mu)\) be a measure space and let \(f_k\in L^1(\mu)\) for \(k\geq1\) satisfy \(\sum_{k=1}^\infty \norm{f_k}_1 < \infty\). Then \(\sum_{k=1}^\infty f_k(x)\) converges absolutely for \(\mu\)-almost every \(x\in X\), its sum \(F\) lies in \(L^1(\mu)\), and \[ \norm{F-\sum_{k=1}^N f_k}_1 \convergesto{N\to\infty} 0, \qquad \int_X F\dmu = \sum_{k=1}^\infty \int_X f_k\dmu . \]