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 $\int_X F(x)\\,d\nu =\int_X F(x)h(x)\\,d\mu$