dourtuntellorvl

2022-06-21

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 $f:\mathsf{X}\to [0,+\mathrm{\infty})$ denoted

$f=\frac{d\mu}{d\lambda}$

satisfying

$\mu (A)={\int}_{A}fd\lambda \phantom{\rule{2em}{0ex}}\mathrm{\forall}A\in \mathcal{X}.$

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 $({\mathbb{R}}_{\ge 0},\mathcal{B}({\mathbb{R}}_{\ge 0}))$. Is this the case that every Radon-Nikodym derivative is a random variable?

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 $f:\mathsf{X}\to [0,+\mathrm{\infty})$ denoted

$f=\frac{d\mu}{d\lambda}$

satisfying

$\mu (A)={\int}_{A}fd\lambda \phantom{\rule{2em}{0ex}}\mathrm{\forall}A\in \mathcal{X}.$

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 $({\mathbb{R}}_{\ge 0},\mathcal{B}({\mathbb{R}}_{\ge 0}))$. Is this the case that every Radon-Nikodym derivative is a random variable?

Braedon Rivas

Beginner2022-06-22Added 24 answers

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, for example, the definition of Chaitin 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.

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.

Craig Mendoza

Beginner2022-06-23Added 6 answers

Short answer: yes.

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