Today we’ll look at a couple corollaries of the Radon-Nikodym chain rule.
First up, we have an analogue of the change of variables formula, which was closely tied to the chain rule in the first place. If and are totally -finite signed measures with , and if is a finite-valued -integrable function, then
which further justifies the the substitution of one “differential measure” for another.
So, define a signed measure as the indefinite integral of . Immediately we know that is totally -finite and that . And, obviously, is the Radon-Nikodym derivative of with respect to . Thus we can invoke the above chain rule to conclude that -a.e. we have
We then know that for every measurable
and the substitution formula follows by putting in for .
Secondly, if and are totally -finite signed measures so that — that is, and — then -a.e. we have
Indeed, , and by definition we have
so serves as the Radon-Nikodym derivative of with respect to itself. Putting this into the chain rule immediately gives us the desired result.