My question follows the logic of Measure Theory by Donald L. Cohn.
Before proving the MCT, the author proves simple versions, i.e. when $f$ and $f_n$ both simple functions.
Proposition 2.3.2. Let $(X, \mathscr{A}, \mu)$ be a measure space, let $f$ belong to $\mathscr{S}_{+}$, and let $\left\{f_n\right\}$ be a nondecreasing sequence of functions in $\mathscr{S}_{+}$such that $f(x)=\lim _n f_n(x)$ holds at each $x$ in $X$. Then $\int f d \mu=\lim _n \int f_n d \mu$.
Proof. It follows from Proposition 2.3.1 that $$ \int f_1 d \mu \leq \int f_2 d \mu \leq \cdots \leq \int f d \mu ; $$ hence $\lim _n \int f_n d \mu$ exists and satisfies $\lim _n \int f_n d \mu \leq \int f d \mu$. We turn to the reverse inequality. Let $\varepsilon$ be a number such that $0<\varepsilon<1$. We will construct a nondecreasing sequence $\left\{g_n\right\}$ of functions in $\mathscr{S}_{+}$such that $g_n \leq f_n$ holds for each $n$ and such that $\lim _n \int g_n d \mu=(1-\varepsilon) \int f d \mu$. Since $\int g_n d \mu \leq \int f_n d \mu$, this will imply that $(1-\varepsilon) \int f d \mu \leq \lim _n \int f_n d \mu$ and, since $\varepsilon$ is arbitrary, that $\int f d \mu \leq \lim _n \int f_n d \mu$. Consequently $\int f d \mu=\lim _n \int f_n d \mu$.
We turn to the construction of the sequence $\left\{g_n\right\}$. Suppose that $a_1, \ldots, a_k$ are the nonzero values of $f$ and that $A_1, \ldots, A_k$ are the sets on which these values occur. Thus $f=\sum_{i=1}^k a_i \chi_{A_i}$. For each $n$ and $i$ let $$ A(n, i)=\left\{x \in A_i: f_n(x) \geq(1-\varepsilon) a_i\right\} . $$
Then each $A(n, i)$ belongs to $\mathscr{A}$, and for each $i$ the sequence $\{A(n, i)\}_{n=1}^{\infty}$ is nondecreasing and satisfies $A_i=\cup_n A(n, i)$. If we let $g_n=\sum_{i=1}^k(1-\varepsilon) a_i \chi_{A(n, i)}$, then $g_n$ belongs to $\mathscr{S}_{+}$and satisfies $g_n \leq f_n$, and we can then conclude the results.
My questions are
the construction of $g_n$ is quite strange to me, do you have any ideas how to come up with the idea to do it this way?
Can we also construct some $g_n$ when proving the Monotone Convergence theorem?
Are there any relations between proving in this way and a more commonly used way (as in the following post)?
syeh_106 (https://math.stackexchange.com/users/235690/syeh-106), Lebesgue's monotone convergence theorem : a question about the proof, URL (version: 2017-05-12): Lebesgue's monotone convergence theorem : a question about the proof