Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Question

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

This is a past qual problem.

I can do this problem for $g$ simple but not in general? Any suggestions for the last step? Thanks

Answer 1

Check this link, it is Radon-Nikodym theorem: http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem

In fact, here $f = \frac{{\rm d}\nu}{{\rm d}\mu}$, and by property 4 in the link the claim is straightforward. Note that the proof of properties are also mentioned there.

Answer 2

For each nonnegative measurable function $g$, there is a sequence of simple functions $g_n \uparrow g$. Now use monotone convergence twice. For the general case, write $g = g^+ - g^-$.