Background
I am self-studying Donald Cohn's Measure Theory second edition. I got stuck on a step of the proof of Theorem 4.5.1. Here is the theorem:
Theorem$\quad$Let $(X,\mathcal{A},\mu)$ be a measure space, let $p$ satisfy $1\leq p<+\infty$, and let $q$ be defined by $\frac{1}{p}+\frac{1}{q}=1$. If $p=1$ and $\mu$ is $\sigma$-finite, or if $1<p<+\infty$ and $\mu$ is arbitrary, then the operator $T$ defined above is an isometric isomorphism of $L^q(X,\mathcal{A},\mu)$ onto $(L^p(X,\mathcal{A},\mu))^*$.
Proof$\quad$ Since we know that $T$ is an isometry (Proposition 3.5.5), we need only show that it is surjective.
Let $F$ be an arbitrary element of $(L^p(X,\mathcal{A},\mu))^*$. First suppose that $\mu(X)<+\infty$ and that $p$ satisfies $1\leq p<+\infty$. We define a function $\nu$ on the $\sigma$-algebra $\mathcal{A}$ by means of the formula $\nu(A)=F(\langle\chi_A\rangle)$. If $\{A_k\}$ is a sequence of disjoint sets in $\mathcal{A}$ and if $A=\bigcup_kA_k$, then the dominated convergence theorem implies that $\lim_{n\to\infty}\left\|\chi_A-\sum_{k=1}^n\chi_{A_k}\right\|_p$. Since $F$ is continuous and linear, this implies that $F(\langle\chi_A\rangle)=\sum_k^{\infty}F(\langle\chi_{A_k}\rangle)$ ....
In this proof, the notation $\langle f\rangle$ is the coset in $L^p(X,\mathscr{A},\mu)$ to which $f$ belongs. In the case of $1\leq p<+\infty$, $\langle f\rangle$ consists of the functions $g$ such that $g=f$ almost everywhere.
In addition, the notation $\chi_A$ stands for the characteristic function of the set $A$.
My Question
Why is $F(\langle\chi_A\rangle)=\sum_k^{\infty}F(\langle\chi_{A_k}\rangle)$?
I think this question shall be broken down to the following two questions:
First, I couldn't see why $\langle\chi_A\rangle=\sum_{k=1}^{\infty}\langle\chi_{A_k}\rangle$ holds.
Second, I couldn't see why $F(\langle\chi_A\rangle)=\sum_k^{\infty}F(\langle\chi_{A_k}\rangle)$? I am confused here because linearity is only defined for finite summation, right? Like $F(ax+by)=aF(x)+bF(y)$. But why can we extend it to an infinite sum?
Could someone please help me with this? Thanks a lot in advance!
