Does the computation of the inverse function of a set that is not in the image of $f$ makes sense?

Question

I am studying measure theory, and I was wondering if the computation of the inverse function $f^{-1}$ of a set that is not in the image of $f$ makes sense. The fact is that all the functions that I have seen in my course are defined $(\Omega,F)\rightarrow (\mathbb{R},B(\mathbb{R}))$. But for example, let us consider the logistic function $f(x)=\frac{1}{1-e^{-x}}$, would that function be defined on $(\mathbb{R},B(\mathbb{R}))\rightarrow ([0,1],B([0,1])$ or in $(\Omega,F)\rightarrow (\mathbb{R},B(\mathbb{R}))$? In the latter case, what would happened for the sets that are not in the image of $f$?.

Answer 1

Note that here $f^{-1}$ does not necessarily refer to the inverse function. Indeed, $f$ might not admit an inverse. Instead, $f^{-1}$ denotes the pre-image under $f$. I.e. $$ f^{-1}(A)=\{x\in\Omega|\;f(x)\in A\} $$ In the case where $A\cap f(\Omega)=\emptyset,$ you'll simply conclude that $f^{-1}(A)=\emptyset$.

Note that if $f$ is injective and $A\subseteq f^{-1}(\Omega)$, then the two notions (pre-images and images under the inverse function) agree and thus, the difference is pedantic at best. However, the pre-image always makes sense for any function and thus, is a better concept in general.