## The Radon-Nikodym Derivative

Okay, so the Radon-Nikodym theorem and its analogue for signed measures tell us that if we have two -finite signed measures and with , then there’s some function so that

But we also know that by definition

If both of these integrals were taken with respect to the same measure, we would know that the equality

for all measurable implies that -almost everywhere. The same thing can’t quite be said here, but it motivates us to say that in some sense we have equality of “differential measures” . In and of itself this doesn’t really make sense, but we define the symbol

and call it the “Radon-Nikodym derivative” of by . Now we can write

The left equality is the Radon-Nikodym theorem, and the right equality is just the substitution of the new symbol for . Of course, this function — and the symbol — is only defined uniquely -almost everywhere.

The notation and name is obviously suggestive of differentiation, and indeed the usual laws of derivatives hold. We’ll start today by the easy property of linearity.

That is, if and are both -finite signed measures, and if and , then is clearly another -finite signed measure. Further, it’s not hard to see if then as well. By the Radon-Nikodym theorem we have functions and so that

for all measurable sets . Then it’s clear that

That is, can serve as the Radon-Nikodym derivative of with respect to . We can also write this in our suggestive notation as

which equation holds -almost everywhere.