I'm quite new to measure theory, and so far I've come across two different definitions for the integral of a real-valued function over a measure space. I'm having trouble showing that the two definitions are equivalent.
Suppose $(X,\Sigma,\mu)$ is a measure space. The first approach starts with defining a simple function on $X$ as any function of the form $g=\sum_{i=0}^n a_i\chi E_i$, where $a_0,a_1,\ldots,a_n$ are constants in $\mathbb{R}$, $E_0,E_1,\ldots,E_n$ are measurable subsets of $X$ with finite measure, and $\chi E_i$ is the characteristic function of $E_i$. For such a function, we define $$\int g =\sum_{i=0}^n a_i\mu E_i.$$ Next, if $f$ is a nonnegative, real-valued function defined on a conegligible/conull subset of $X$, we say that $f$ is integrable if there is a non-decreasing sequence $\langle f_n\rangle_{n\in\mathbb{N}}$ of non-negative simple functions such that $\sup_{n\in\mathbb{N}}\int f_n<\infty$ and $\lim_{n\to\infty}f_n =f$ almost everywhere. In this case, we set $$\int f = \sup\left\{\int g:g\text{ is a simple function and }g\leq f\text{ almost everywhere}\right\}.$$ Finally, if $f$ is an arbitrary real-valued function defined on a conegligible subset of $X$, we say that $f$ is integrable if we can write $f=f_1-f_2$ where $f_1$ and $f_2$ are nonnegative, integrable functions (in the sense defined earlier). In this case, we set $\int f= \int f_1- \int f_2$.
For the second approach, we start by defining a quasi-simple function on $X$ as any function of the form $g=\sum_{i=0}^{\color{red}\infty} a_i \chi G_i$ where $\langle G_i\rangle_{i\in\color{red}{\mathbb{N}}}$ is a $\color{red}{\text{partition}}$ of $X$ into measurable sets, $\langle a_i\rangle_{i\in\color{red}{\mathbb{N}}}$ is a sequence in $\mathbb{R}$, and $$\sum_{i=0}^\infty |a_i|\mu G_i<\infty.$$ In the above sum, we count $0\cdot \infty$ as $0$, so that $G_i$ can have infinite measure so long as $a_i=0$. For such a function, we define $$\int g=\sum_{i=0}^\infty a_i\mu G_i.$$ We say a real-valued function $f$ defined on a conegligible subset of $X$ is integrable if the two values $$\sup\left\{\int g:g\text{ is a quasi-simple function and }g\leq f\text{ almost everywhere}\right\}$$ $$\inf\left\{\int h:h\text{ is a quasi-simple function and }h\geq f\text{ almost everywhere}\right\}$$ are equal, in which case we set $\int f$ equal to the common value.
I noticed that the difference in approach between these two definitions becomes more apparent when considering the case in which $f$ is unbounded. Indeed, if we suppose $f$ is unbounded below, then although there are quasi-simple functions which are less than $f$, there are no simple functions less than $f$ (so that splitting up $f$ into $f_1-f_2$ as done in the first definition is necessary).
I would greatly appreciate any help in showing that these two definitions are equivalent. For further context, showing the equivalence of these two definitions is exercise 122Yd in Fremlin's first volume of Measure Theory.