I hope this question hasn't already been posted before in some other form (I couldn't find it, so if it has, please pardon me). I found this question on an old qualifier, but I am completely lost as to how to even begin. My only thoughts at this point are to somehow construct $h$ from some nicer dense functions, but I am not sure how to proceed even with that.
If you have any hints, thoughts, suggestions, or possible solutions, I would welcome your input.
Here is how it appears (verbatim):
"Let $(X,M,\mu)$ be a measure space and let $T:X \to X$ be measurable, such that, if $E \in M$, $\mu(E)=0$ then $\mu(T^{-1}(E))=0$. Prove that there exists $h \in L^{1}(\mu)$ such that
$$ \int f \circ T \, d\mu = \int fh \, d\mu$$
for all $f \in L^{\infty}(\mu)$."
Thank you for your time.
Hint: try the Radon-Nikodym theorem.