This is a follow up post to my previous here. I am piecing parts of the Completeness in $L_p$ proof. I have labelled parts I am unsure of $(1),(2)$ and $(3)$. I have argued why they are true. However, it was messy as I had to check each condition of DCT.
I hope one could check my argument or give a clean argument to the three parts.
We now define $f$ on $X$ by $$ f(x) \quad \quad \quad \quad \quad = g_1(x) + \sum_{k=1}^{\infty} (g_{k+1}(x) -g_k(x) ), \quad x \in E, $$ $$ = 0 , \quad \quad x \notin E. $$ Since $|g_k| \le |g_1| + \sum_{j=1}^{k-1} |g_{j+1} -g_j| \le g $ and since $\color{blue}{\text{converges $(g_k)$ a.e. to $f$} \quad (1) }$, $\color{blue}{\text{the Dominated Convergence Theorem implies that $f \in L_p$.} \quad \quad (2)}$
Since $|f- g_k|^p \le 2^pg^p,$ $ \color{blue}{\text{ we infer from the DCT that $0 = \lim ||f -g_k||_p$, so that $(g_k)$ converges in $L_p$ to $f$. } \quad \quad (3) }$
If $m \ge M(\varepsilon)$ and $k$ is sufficiently large, then $$ \int |f_m - g_k|^p \, d \mu < \varepsilon^p . $$ Apply Fatou's Lemma to conclude that $$ \int |f_m - f|^p \, d \mu \le \lim \inf _{k \rightarrow \infty} \int | f_m - g_k| ^p \, d \mu \le \varepsilon^p.$$ whenever $m \ge M(\varepsilon)$. This proves that the sequence $(f_n)$ converges to $f$ in the norm of $L_p$.
How I would argue
(1): As $g(x) := |g_1(x)| + \sum_{k=1}^{\infty} |g_{k+1}(x) -g_k(x)| <+\infty $ on $E$, we know the series converges absolutely, hence the series in $f$ converges on $E$. So $(g_k)$ converges a.e. to $f$ on $E$.
(2): By continuity $|g_k|^p$ converges a.e. to $|f|^p$ on $E$. We know that by definition $g_k \in L_p$ so $\int |g_k|^p \, d \mu < \infty$ and $|g_k|^p$ is integrable.
It has been shown that $|g_k|^p \le |g|^p $ where $|g|^p$ is integrable. Hence, the conditions of DCT are satisfied, we then obtain,
$$ \int |f|^p \, d \mu \stackrel{DCT}{=} \lim \int |g_k|^p \, d \mu \le \lim \int |g|^p \, d \mu < + \infty .$$
Since $f$ is limit of measurable functions, it is measurable. So $f \in L_p$.
(3): Apply DCT on the sequence of functions: $(|f-g_k|^p)$. We first note that they are integrable, since $|f-g_k|$ is measurable and $\phi(x) = x^p$ is a continuous map, $\phi(f-g_k) = |f-g_k|^p$ is measurable.
Also, as $|f-g_k|^p \le 2^p g^p$, and the RHS is integrable, we deduce that $|f-g_k|^p$ is integrable. We have previously shown that $|f-g_k|^p \rightarrow 0 $ a.e. So by DCT, $$ 0 \stackrel{DCT}{=} \lim \int \, |f-g_k|^p \, d \mu = \lim ||f - g_k ||_p^p $$ But as $\phi(x)=x^p$ is continuous, we deduce $$ \lim ||f-g_k||_p = 0 .$$
Summary:
I would like to know how one writes a clean argument in this case, and why the author states as though all conditions of DCT are trivially satisfied.
Thanks in advance.
Lebesgue Dominated Convergence Theorem. Let $(f_n)$ be a sequence of integrable functions which converges almost everywhere to a real valued measurable function $f$. If there exists an integrable function $g$ such that $|f_n| \le g$ for al $n$, then $f$ is integrable and $$ \int f \, d \mu = \lim \int f_n \, d \mu .$$
PS: How do I append "hide" bars. As I would like to include the whole proof, but it would take up the whole page.