Recall that Fubini's theorem in a very basic form says that if $(X,\mu)$ and $(Y,\lambda)$ are $\sigma$-finite positive measure spaces and $f$ is a positive function on $X \times Y$ that is measurable with respect to the product measure, then $$\int_X (\int_Y f(x,y) d \lambda(y)) d\mu(x) = \int_Y (\int_X f(x,y) d \mu(x)) d \lambda(y).$$
I am curious, what happens in Fubini's theorem if we remove the requirement that the measures are positive, and instead assume that they are complex Borel measures? Does some version of Fubini's theorem still hold? If not, can we put more requirements on either $f$ or the measures to make it work for complex Borel measures? My guess would be that it would be good to look at the Radon-Nikodym theorem to show that something similar holds, but I am not sure.