TITLE: Measure Theory - Problem with definition about simple functions
QUESTION [5 upvotes]: I did this question lately and then realized what my mistake was. I got a good help!
But looks like only now I understand what was the real problem, to begin with. And this brought me to the same questioning. 
In the same (same used in the previous question) book of Measure theory and Integration of Folland, we have the following definition.

The definition of integration is made only for simple function with standard representation, and this is with a good reason, the text above says why. Therefore, to the purposes of integration, it is sufficient to consider only simple functions with standard representation in the definition of integral in $L^+$ ?
With this in mind, why couldn't we just consider $$\int f = \sup\Big\{ \int \Phi: \ 0\leq\Phi\leq f, \Phi \textrm{ is a simple and in standard representation}  \Big\}$$
instead $$\int f = \sup\Big\{ \int \Phi: \ 0\leq\Phi\leq f, \Phi \textrm{ is a simple}  \Big\}?$$
The last one is the "official" definition.
If we accept this, the answers I was given are wrong. So, I want (and need, this is too fundamental to just let alone) to know what is wrong with my reasoning.
Thank you!

REPLY [4 votes]: The point that I think you're missing is that no matter how you represent a simple function, it's integral will be the same. This justifies the fact that it doesn't matter if the simple function is in its standard representation or not. That is, integrating a simple function is well-defined. The reason that this is even stressed is because there are infinitely many ways to represent the $\textit{same}$ simple function; so we need the integral to be the same no matter how it's represented. For example, if $\phi(x) = \chi_{(0,1]}(x) + \chi_{(1,2]}(x) = \chi_{(0,2]}(x)$, then the integrals (with respect to Lebesgue measure on $\mathbb R$) are the same:
$$\int \phi(x) dm = m((0,1]) + m((1,2]) = 1 + 1 = 2 = m((0,2]) = \int \phi(x) dm$$
Putting a simple function in its standard representation is useful when you're proving things for simple functions. For instance, when proving the integral is additive for simple functions, it is best that the sets $E_i$ are disjoint.