Why is $\int g^p<\infty$ in this proof of completeness of $L^p$?

Question

I am trying to understand a step in the proof of completeness of $L^p$ in Stein-Shakarchi's Functional Analysis. (See the proof on page 5 of the link or at the end of this post.)

In the first part of the proof, it is shown that $$ \int g^p<\infty. $$ Question: how is this done?

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My thought was that as the partial sum $_()$ converges, the whole series $g=|g|$ converges which can be seen as $|g|<\infty$ thus $|g|^p<\infty$, and thus $\int|g|^p<\infty$. However, such argument will get into conflict with "$g$ converges almost everywhere." which is in this answer to another question of mine.

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Answer 1

In the proof, it is shown that $$ \|S_K(g)\|_{L^p}\leq \|f_{n_1}\|_{L^p}+\sum_{k=1}^K2^{-k}, $$ which implies that (it is an instructive exercise to show that instead of $\limsup_{K\to\infty}$, one actually has $\lim_{K\to\infty}$ in (*)) $$ \lim_{K\to\infty} \|S_K(g)\|_{L^p} \leq \|f_{n_1}\|_{L^p}+1=:C\tag{*} $$ for some positive real constant $C$. By continuity of the function $h\mapsto h^{p}$ on $[0,\infty)$ and the definition of the $L^p$ norm, we have $$ \lim_{K\to\infty}\int |S_K(g)|^p\leq C^p.\tag{1} $$

(Remark: $S_K(g)$ is nonnegative for all $K$ and thus $|S_K(g)|=S_K(g)$.)

On the other hand, $S_K(g)\to g$ everywhere by the definitions of $S_K(g)$ and $g$. By the monotone convergence theorem, $$ \lim_{K\to\infty}\int[S_K(g)(x)]^p\,dx=\int\lim_{K\to\infty}[S_K(g)(x)]^p\,dx =\int g(x)^p\,dx\tag{2} $$ Combining (1) and (2) together, we have $$ \int g^p<\infty. $$

Answer 2

The integral is finite because we can take limits on either side of $$ ||S_K(g)||_p^p\leq (||f_{n_1}||+\sum_{k=1}^K 2^{-k})^p $$ and using monotone convergence to move the limit inside the norm on the left hand side $$ \lim_{K\to \infty}||S_K(g)||_p^p\leq (||f_{n_1}||+\sum_{k=1}^\infty 2^{-k})^p=(||f_{n_1}||+1)^p\\ \stackrel{\text{monotone convergence}}\implies ||\lim_{K\to \infty}S_K(g)||_p^p\stackrel{\text{definition of g}}=||g||_p^p\leq (||f_{n_1}||+1)^p $$