I am self-studying measure theory, and I got some trouble understanding the construction of the integral. Here is the first two stages of the construction:

Stage 1$\quad$ We begin with the simple function. Let $(X,\mathscr{A})$ be a measurable space. We will denote by $\mathscr{S}$ the collection of all simple real-valued $\mathscr{A}$-measurable functions on $X$ and by $\mathscr{S}_+$ the collection of nonnegative functions in $\mathscr{S}$.

Let $\mu$ be a measure on $(X,\mathscr{A})$. If $f$ belongs to $\mathscr{S}_+$ and is given by $f = \sum_{i=1}^ma_i\chi_{A_i}$ where $a_1,\dots,a_m$ are nonnegative real numbers and $A_1,\dots,A_m$ are disjoint subsets of $X$ that belong to $\mathscr{A}$, then $\int f d\mu$, the integral of $f$ with respect to $\mu$, is defined to be $\sum_{i=1}^ma_i\mu(A_i)$ (note that this sum is either a nonnegative real number or $+\infty$).

Stage 2$\quad$ As our next step, we define the integral of an arbitrary $[0,+\infty]$-valued $\mathscr{A}$-measurable function on $X$. For such a function $f$, let \begin{align*} \int fd\mu = \sup\left\{\int gd\mu:g\in\mathscr{S}_+\ \text{and}\ g\leq f\right\}.\tag1 \end{align*}

(There is a final stage 3, but my problem is with the stage 2.)

My question is this: How is the equation (1) compatible with the summation definition of $f$ in Stage 1?

I can see that, if $f$ and $g$ both belong to $\mathscr{S}_+$ and if $f(x)\leq g(x)$ holds at each $x$ in $X$, then $\int fd\mu \leq \int gd\mu$. But I am not quite sure how the two definitions agree with each other. Could someone please help me out? Thanks a lot in advance!

Reference: Measure Theory by Donald Cohn page 55.

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    $\begingroup$ Definition in summation notation is defined only for non-negative real valued simple functions. Definition in $(1)$ is general non-negative measurable functions. See that simple functions are measurable and that definition in summation is included in the definition in $(1)$. $\endgroup$ Apr 15 at 17:35
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    $\begingroup$ Just an informal hint: Please try to draw graphs of simple functions and try to understand and figure what these expressions in summation actually mean in the graphs. That way you will get a clear idea of the machinery behind the symbols. Even my professor encouraged the students to do the same to gain a proper insight into what was happening. $\endgroup$ Apr 15 at 17:39


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