Im trying to prove a generalization of the Radon Nykodym theorem, but im having troubles even for finite measures, could someone help?


Let $\mu$ and $\nu$ two $\sigma$-finite measures in $\(X,\mathcal{F})$. If $\nu$ << $\mu$, then there exists a non-negative function $h \in L^1(X,\mu)$, such that for every function $F\in M^+(X,\mathcal{F})$, it is satisfied that $\int\_X F(x)\\,d\nu =\int\_X F(x)h(x)\\,d\mu$ 



About a generalization of the Radon Nikodym Theorem
If you assume you have proven the Radon-Nikodym theorem for measures $\mu,\nu$, $\nu <<\mu$, then for any $F$ in such a class, the measure $\mu\_F,\nu\_F$ defined by 


$\mu\_F(E):=\int\_E F(x)\;d\mu$,


$\nu\_F(E):=\int\_E F(x)\;d\nu$, 


then since $\nu\_F << \mu\_F$ (if your definition of absolute continuity is equivalent to $\nu(E)=0$ whenever $\mu(E)=0$), you can apply the Radon-Nikodym theorem for $\nu\_F,\mu\_F$ and achieve the desired result.


