Let $(\mathsf{X}, \mathcal{X}, \mu)$ be a measure space with $\mu$ begin $\sigma$-finite.
Definition of Random Variable: Let $(\mathsf{Y}, \mathcal{Y})$ be a measurable space. A function $\xi:\mathsf{X}\to\mathsf{Y}$ is a random variable if $\xi^{-1}(A)\in\mathcal{X}$ for every $A\in\mathcal{Y}$. We say that the random variable is a $\mathcal{X}$-measurable function.
Definition of Radon-Nikodym derivative: Let $\lambda$ be $\sigma$-finite measure with $\mu \ll \lambda$. Then there exists a $\mathcal{X}$-measurable function $f:\mathsf{X}\to[0, +\infty)$ denoted
$$
f = \frac{d\mu}{d\lambda}
$$
satisfying
$$
\mu(A) = \int_A f d\lambda \qquad \forall A\in\mathcal{X}.
$$
It seems that the Radon-Nikodym derivative is a $\mathcal{X}$-measurable function so by definition it should be a random variable between $(\mathsf{X}, \mathcal{X})$ and $(\mathbb{R}_{\geq 0}, \mathcal{B}(\mathbb{R}_{\geq 0}))$. Is this the case that every Radon-Nikodym derivative is a random variable?
Best Answer
Short answer: yes.
Long answer: the axiomatization of probability by Kolmogorov begins by giving suggestive names to familiar objects. For example, a mere "measurable function" is called "random variable". Of course, a "random variable" isn't random at all (by the way, to my knowledge, the word "random" has formal definitions only in some areas of math - see, for example, the definition of Chaitin, as pointed out by @postmortes in the comments, that is more or less related to the "unpredictable" character of randomness - none of which are directly related to probability theory). So, any map between finite sets is a random variable, if one assumes that the finite sets are given the discrete $\sigma$-algebra. For example, the map sending each human being to its height in centimeters is a random variable. The identity map, from $\{4,7,56\}$ to itself, is also a random variable.
Now, probability theory, as a theory with fancy names, has proved to be very successful in modelling "true random events in real life", but this is another story; to understand why is a problem of dynamical systems, physics, maybe philosophy. But if it is just thought a branch of measure theory, it is just like any other mathematical theory.