Abstract integral - Borel measures - $L^p$ spaces 
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.

Let $\nu\colon\mathcal M\to\Bbb R$ given by $\nu(S):=\mu(T^{-1}S)$. We can show that this defines a measure. By assumption, this one is absolutely continuous to $\nu$. Radon-Nikodym theorem gives $h$.
