Prove that there exists $h \in L^1(\mu)$ such that $h \geq 0$ and $\int f \circ T = \int fh$

Question

I am studying for the qualifiers and am trying to solve the following question:

Let $(X, \mu, \mathscr{M})$ be a finite measure space. Suppose $T: X \to X$ is measurable and $\mu(T^{-1}(E))=0$ whenever $E \in \mathscr{M}$ and $\mu(E)=0$. Prove that there exists $h \in L^1(\mu)$ such that $h \geq 0$ and $\int f \circ T \, \rm{d}\mu= \int fh \, \rm{d}\mu$ for all $f \in L^\infty (\mu)$.

I am not sure where to even start with this problem. Any advice on where to begin would be appreciated.

Answer 1

The pullback measure satisfied the Radon-Nikodym condition. Invoke the Radon-Nikodym theorem to get existence of the function.