I got lost when reading the proof of $L^{\infty}$ is complete. The book proceed the proof as follows:
We show that each absolutely convergent series in $L^{\infty}(X,\mathscr{A},\mu)$ is convergent. We do this by considering functions (instead of equivalence classes) in $\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$.
Let $\{f_k\}$ be a sequence of functions that belong to $\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ and satisfy $\sum_k\|f_k\|<+\infty$. For each positive integer $k$, let $N_k=\{x\in X:|f_k(x)|>\|f_k\|_{\infty}\}$. Then the series $\sum_kf_k(x)$ converges at each $x$ outside $\bigcup_kN_k$, and the function $f$ defined by\begin{align*}f(x) = \begin{cases}\sum_kf_k(x)\quad&\text{if $x\notin\bigcup_kN_k$},\\\\0\quad&\text{if $x\in\bigcup_kN_k$}\end{cases}\end{align*}is bounded and $\mathscr{A}$-measurable. Since $\bigcup_kN_k$ is locally $\mu$-null, the inequality\begin{align*}\left\|f-\sum_{k=1}^nf_k\right\|_{\infty} \leq \sum_{k=n+1}^{\infty}\|f_k\|_{\infty}\tag1\end{align*}holds for each $n$, and so\begin{align*}\lim_{n\to\infty}\left\|f-\sum_{k=1}^nf_k\right\|_{\infty} \leq \lim_{n\to\infty}\sum_{k=n+1}^{\infty}\|f_k\|_{\infty} = 0.\tag2\end{align*}Thus $L^{\infty}(X,\mathscr{A},\mu)$ is complete.
I have a couple of questions about this proof.
The definition of $L^p(X,\mathscr{A},\mu)$ says that the elements of $L^p(X,\mathscr{A},\mu)$ are equivalence classes of functions. Why is it legit to proceed the proof by considering functions in $\mathscr{L}^p(X,\mathscr{A},\mu)$?
In the proof, it says "the series $\sum_kf_k(x)$ converges at each $x$ outside $\bigcup_kN_k$". Here is how I understand this step, and I want to know if it is correct?
If $x\notin\bigcup_{k=1}^{\infty}N_k$, then $|f_k(x)|\leq\|f_k\|_{\infty}$ for all $k\in\mathbb{N}$, and thus the convergence of $\sum_{k=1}^{\infty}\|f_k\|_{\infty}$ implies that $\sum_{k=1}^{\infty}f_k(x)$ converges by the comparison test.
The proof claims that $f$ is $\mathscr{A}$-measurable. I couldn't see why this is true. I want to show that for each $t\in\mathbb{R}$ the set $\{x\in X:f(x)<t\}\in\mathscr{A}$. But I honestly don't know how to do this. Can someone please help me out?
I got complete lost by inequality (1) and (2). Why does $\bigcup_kN_k$ being locally $\mu$-null imply that the inequality (1) holds for each $n$? Why can we just take limit for both side without proving the limits exist?
Please please help! I really appreciate it!
Note:$\quad$ The book I am reading defines $\|f\|_{\infty}$ to be the infimum of those nonnegative numbers $M$ such that $\{x\in X:|f(x)>M|\}$ is locally $\mu$-null.