$\dfrac{d \rho }{d\mu} = \dfrac{d\rho} {d\nu}\cdot\dfrac{d\nu} {d\mu},$ where $\nu, \mu$ and $\rho$ are finite signed measures.

Question

There are not any other assumptions, so is it necessary for $\mu,\nu$ to be $\sigma$-finite?

Suppose $\nu, \mu$ and $\rho$ are finite signed measures, $\nu\ll \mu$ and $\rho \ll \nu$. Then

$\dfrac{d \rho }{d\mu} = \dfrac{d\rho} {d\nu}\cdot\dfrac{d\nu} {d\mu}.$

Answer 1

The measures don't have to be finite, but they should all be $\sigma$-finite for all derivatives to exists.