What do we mean when we say that a function $f$ takes the value $ \infty $? What do we mean when we say that a function $f$ takes the value $ \infty $?
In measure theory it is common to let mappings take values in the extended real number system.
But still it doesn't make sense to say that $f(x)= \infty $ for any specific $x $. So what is meant with $f : X \mapsto [-\infty, \infty ]$; or the preimage of the set whwer $f =\infty $.

I can see two possibilities, for instance we may say $1/x =\infty $ at $x=0 $. Or we could say $f= \infty $ on some set $E $ if $f $ is unbounded here.
But please help me with clarifying the matter.
Thanks in advance!
 A: In measure theory, we just allow $\infty$ as a value, because it is convenient to do so.
For example, let $f : \Bbb{R} \to \Bbb{R}$ be a (bounded) function with $\int |f|\, dx < \infty$.
Then, we can just define (without having to worry about convergence questions)
$$
g : \Bbb{R} \to [0,\infty], x \mapsto \sum_{n \in \Bbb{Z}} |f(x+n)|.
$$
By usual Theorems like monotone convergence, we then see that
$$
\int_0^1 g(x) \,dx = \sum_{n} \int_0^1 |f(x+n)| \, dx = \sum_n \int_n^{n+1} |f(y)|\, dy = \int_\Bbb{R} |f|\, dx <\infty.
$$
Now, we can conclude a posteriori that $g(x) < \infty$ almost everywhere (and thus the sum converges absolutely almost everywhere).
If we had not allowed functions to possibly take the value $\infty$, we would have had to artificially restrict our assumptions on $f$, or to change the summation in $g$ to a finite summation, ...
Hence, the argument was tremendously simplified by allowing $g$ to take the value $\infty$.
In this case, this had nothing to do with something like
$$
\frac{1}{x} \to \infty \text{ for } x \downarrow 0.
$$
In these cases, the concept of "taking the value $\infty$" is also not too useful, because if you want to integrate such a function, you can just as well define
$$
f:x\mapsto\begin{cases}
\frac{1}{x}, & x\neq0,\\
\sqrt{\pi}, & x=0,
\end{cases}
$$
because the singleton $\{0\}$ is a null-set and thus irrelevant for integration purposes. (We could also have used $\infty$ or $0$ instead of $\sqrt{\pi}$, but as I said this is not the most important application of "taking the value $\infty$").
