For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$
I could prove that $\sigma(f^{-1}(C)) \subset f^{-1}(\sigma(C))$ since complements and unions are 'preserved' by function inverse. But how do I go the other way?
EDIT: One way to go the other way would be to argue that any set in $\sigma(C)$ must be built by repeatedly applying the complement, union and intersection operations to elements of $C$ and all these operations are preserved when taking the inverse. The problem I am facing with the approach is formalizing the word "repeatedly".
[not-homework]