Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h \ge 0$ and $$\int f \circ T \, d\mu = \int fh \, d\mu,$$ for all $\, f \in L^\infty(\mu)$.
What I believe I understand about this problem is to take a simple function such as $f= \chi_E$, use linear combinations, and then an approximation. Please help in this solution.
