A very important measure. Let $f\in L^1(X,\mathcal{A},\mu)$; we define an application $\nu\colon\mathcal{A}\to \mathbb{R}$ placing $$\large \nu(E):=\int_E f\;d\mu\quad (E\in\mathcal{A}).$$
Let $\{E_k\}_k\subseteq \mathcal{A}$ be a sequence of disjoint sets and define $E=\bigcup_{k=1}^\infty E_k$.
Claim. $\large \nu(E)=\sum_{k=1}^\infty\nu(E_k)$
For definition, $\nu(E)=\nu_+(E)-\nu_-(E)$, where $\nu_\pm\colon\mathcal{A}\to\mathbb{R}_+$ and $\large \nu_\pm=\int_E f_\pm\; d\mu$, since $\nu_\pm$ are measures we have
$$
\begin{aligned}
\large \nu(E)=\nu_+(E)-\nu_-(E)&=\left(\sum_{k=1}^\infty\nu_+(E_k)\right)-\left(\sum_{k=1}^\infty\nu_-(E_k)\right)\\
&\color{RED}{=}\sum_{k=1}^\infty \left(\nu_+(E_k)-\nu_-(E_k)\right)\\
&=\sum_{k=1}^\infty \nu(E_k)
\end{aligned}
$$
I know that the red equality derives from the fact that the series $$\large \sum_{k=1}^\infty \nu_\pm(E_k)=\int_E f_\pm\,d\mu<\infty.$$

My text also states that in addition to series convergence, known theorems on absolutely convergent series have been used. However, I do not understand where these theorems have been used. Could you give me some explanation about it? Thanks.

 A: Actually for emphasis and logical flow,  I'd re-order and place the red equality as: $$\begin{align}\tag{1}\sum_{k\ge1}\nu(E_k)&=\sum_{k\ge1}(\nu_+(E_k)-\nu_-(E_k))\\\tag{2}&\color{red}{=}\sum_{k\ge1}\nu_+(E_k)-\sum_{k\ge1}\nu_-(E_k)\\&=\nu_+(E)-\nu_-(E)\\&=\nu(E)\end{align}$$Since, given convergence of $2)$, convergence and equality with $1)$ is automatic from basic limit properties, but given convergence of $1)$ it is not always possible to obtain convergence and/or equality with $2)$.
The classic example of this is $\nu_{\pm}(E_k):=k$ for all $k$. Then the sum in $1)$ converges to $0$ as all its summands are $0$, but sum $2)$ is ill-defined. However, since the integrability of $f$ implies both series in $2)$ are convergent, you get equality with $1)$ by limit properties.
I don't think the transition $1)\to2)$ requires absolute convergence of the series, but comments about absolute convergence are often referring to the beautiful phenomenon described by the Riemann series theorem where conditionally-but-not-absolutely convergent series can be made to attain any arbitrary extended real value (!) via reordering of the terms.
N.B. The series $1)$ is absolutely convergent anyway, since: $$\sum_{k\ge1}|\nu_+(E_k)-\nu_-(E_k)|=\sum_{k\ge1}|\nu(E_k)|\le\int_E|f|\,\mathrm{d}\mu\lt\infty$$
