When one develops the theory of integration, why is it usually the case that simple functions are not allowed to take the value $\infty$?
Recall that if $(X,\mathcal{M})$ is a measureable space then a simple function is a measurable $s:X\rightarrow [0,\infty)$ (here $[0,\infty)$ is endowed with its usual subspace topology which generates its Borel sets $\mathcal{B}[0,\infty)$) if $s$ can be written as a finite combination of indicators of sets in $\mathcal{M}$ with coefficients in $[0,\infty)$: $$s=\sum_{i=1}^n \alpha_i \mathbf{1}_{A_i},$$where $\alpha_i\in[0,\infty)$ for all $i=1,\dots, n$ and $A_i\in\mathcal{M}$ for all $i=1,\dots, n$.
Since integration theory is built to handle extended real-valued measurable functions, it seems silly that one not allow simple functions to have a coefficient of $\infty$.
As an example, let $(X,\mathcal{M}, \mu)$ be a measure space and define the function $f(x) = \infty$ for all $x\in X$. This is a $(\mathcal{B}[0,\infty], \mathcal{M})$-measurable function, and it would seem to make sense to define $\int_X f\,d\mu = \infty\cdot \mu(X)$, since $f = \infty\cdot \mathbf{1}_X$, and therefore consider $f$ as a simple function. Of course, one can get the same answer by considering a sequence of simple functions (whose range is $[0,\infty)$) converging pointwise to $f$ and then taking the definition of $\int_X f\,d\mu$ to be the usual one, i.e., the supremum of the integrals of all simple functions less than $f$.
Long story short, it doesn't seem like we run into any problems if we allow simple functions to have $\infty$ as one of its coefficients. The best reason I can come up with as to why books (Rudin, Folland, etc.) are adamant about defining simple functions as not taking value $\infty$ is that it is not necessary to define the integral for "simple functions" that take value $\infty$; rather one can recover the integral of such functions by the "supremum definition" of the integral. But then again, this is simply a definition too, so is there any more generality in one approach versus the other?
By the way, this previous post does not give a satisfying answer to this particular question.