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

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I'm afraid your question is not very clear. At least I don't really understand what you are having difficulties or what you are asking. Try to give some more context and more details. –  Ittay Weiss Nov 25 '12 at 7:33

Take a approximation of F by simple functions. Then it is a consequence of the (usual) Radon-Nikodym.

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