I am trying to understand measure theory and I have encountered the following question:
Let $\Omega=(1,2,3,4)$, $F=\{ \{ \emptyset\}\{ 1\} \{ 3\} \{ 1,3\}\{ 2,4\} \{ 1,2,4\} \{2,3,4\} \{ \Omega\} \}$,$ H=\{ \{ \emptyset\}\{ 1\} \{ 4\} \{ 1,4\}\{ 2,3\} \{ 1,2,3\} \{2,3,4\} \{ \Omega\} \}$
We have the following function map $f:\Omega -> \mathbb{R} $ such that $f=(-1)^n$. Decide whether f is measurable or not with respect to F and H.
To do this I need to find the inverse function of f and see if $f^{-1}(A)\in F $ For all $A \in H$ (and then the opposite case).
I get the inverse function to be $f^{-1}(A)=-ln(A)$ which does not make sence since the logarithm of and integer is not likely to be an integer and therefore the preimage is pretty much never going to belong to F or H.
Any help would be massively apprichiated.