Can simple functions take the value infinity?

Question

I don't think my book is clear about this. It is "a course in real analysis", by weiss.

Now I am in the chapter about the general lebesge integral, and we are going to develop the non-negative lebesgue integral for abstract measurable sets. They define simple functions like this:

As you can see,t hey don't mention if the a values can be infinity?

Afterwords they have this proposition:

Notice that they now are talking about the extended real valued functions. So the function can be infinity.

Now there are two possibilites as far as I see.

1. The simple functions can take the value infinity, and hence what the proposition gives is true, since then it can converge to infinity.

2. The simple function can only take values in the real numbers, but since they are arbitrary big, they can still converge to infinity in the sense that they can grow beyond all bounds.

Is one of these correct, if so, which? It may be that none is correct and they mean something else?

Answer 1

I don't have a book nearby, but usually simple functions are not allowed to take infinite values. (That is, your option 2 is correct). If you inspect the proof of Proposition 4.7, you will likely find that the functions they construct attain only finite values.

(By the way, one such construction is: let $s_n(x)$ be the largest number not exceeding $f(x)$ whose decimal representation has at most $n$ digits.)

In Real Analysis by Folland I see an explicit statement that simple functions are not allowed to attain infinite values. After all, we want simple functions to be indeed simple, to make it easier to check various integral properties for them, and then pass to $L^1$ (the space of integrable functions) by density.

Answer 2

I don't think there is a contradiction here. Simple functions are not allowed to take infinity as a value, however the sequence of simple functions still can converge to the extended function.