# Is every radon-nikodym derivative a random variable?

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?

• Random variable is a measurable function on a probability space (i.e. $\mu(X) = 1$), not any measure space.
– ajr
Commented Mar 24, 2022 at 10:59
• @ajr Ah! I have never noticed that! Commented Mar 24, 2022 at 11:03

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.