If , , and are totally -finite signed measures so that and , then -a.e. we have
By the linearity we showed last time, if this holds for the upper and lower variations of then it holds for itself, and so we may assume that is also a measure. We can further simplify by using Hahn decompositions with respect to both and , passing to subspaces on which each of our signed measures has a constant sign. We will from here on assume that and are (positive) measures, and the case where one (or the other, or both) has a constant negative sign has a similar proof.
Let’s also simplify things by writing
Since and are both non-negative there is also no loss of generality in assuming that and are everywhere non-negative.
for every measurable . For every measurable set we find that
and so for all the simple we conclude that
Passing to the limit, we find that
and so the product serves as the Radon-Nikodym derivative of in terms of , and it’s uniquely defined -almost everywhere.