Here is statement of the theorem and I am stuck on the very last line of the following proof:
I can not see by Radon Nikodym, we have part 1 and 2 follow right away. By Radon Nikodym, we have some $g \in L^1$ such that $\frac{d\lambda}{d\mu} = g$. If we set $f' = g$, then I see $f' = g$ is in $L^1$ by Radon Nikodym. But what tells us the Radon Nikodym derivative $g$ is the classical derivative? This question might be related to the following theorem which was proved ahead:
But I am not sure how we can apply this theorem here.