Consider any two continuous random variables defined on the Borel-measurable sets on $\mathbb{R}$.
I would like to reference a proof of the following statement:
"The distributions of the two random variables are related by a Radon-Nikodym derivative, and the derivative is the quotient of their PDFs".
I am looking for a text-book where the result is proven for the general case of any two continuous RVs.
Let $f$ and $g$ be the PDFs of two random variables with respect to the Lebesgue measure. Then, we may express them as Radon-Nikodym derivatives:
$$f = \frac{d \mu}{d \lambda} \qquad g = \frac{d \nu}{d \lambda}$$
where $\lambda$ is the Lebesgue measure and $\mu$ and $\nu$ are two probability measures. The density of $\mu$ with respect to $\nu$ is then given by the chain rule for RN-derivatives: $$\frac{f}{g} = \frac{\frac{d \mu}{d \lambda}}{\frac{d \nu}{d \lambda}} = \frac{d \mu}{d \nu}$$
A proof of the chain rule for RN derivatives can be found in Folland's Real Analysis, Proposition 3.9(b).