When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$ be some probability measure over this space which we refer to as a distribution of some random variable. The usual definition states that there is **some** probability space $(\Omega,\mathscr F,\mathsf P)$, the random variable is
$$
 \xi:(\Omega,\mathscr F)\to(X,\mathscr A)
$$
i.e. it is a measurable map, and its distribution is a pushforward measure: 
$$
\mathsf Q:=\xi\_\*(\mathsf P)
$$
i.e. $\mathsf Q(A) = \mathsf P(\xi^{-1}(A))$ for any $A\in \mathscr A$.


Clearly, given $(X,\mathscr A,\mathsf Q)$ for a single random variable there is no reason to come up with a new sample space and we can take $(\Omega,\mathscr F,\mathsf P) = (X,\mathscr A,\mathsf Q)$ and $\xi:=\mathrm{id}\_X$. 


Let us stick to this latter case. It may happen, that there is a map 
$$
 \eta:(X,\mathscr A)\to(X,\mathscr A)
$$
such that $\eta\neq \rm id\_X$ but still it holds that $\mathsf Q = \eta\_\*(\mathsf Q)$. I wonder if the existence of this other maps is studied somewhere.




---


The brief statement of the problem is thus the following: given a probability space $(X,\mathscr A,\mathsf Q)$ if the identity map $\rm id\_X$ is the unique solution of the equation
$$
 \mathsf Q = \xi\_\*(\mathsf Q) \tag{1}
$$
where the variable $\xi$ is any measurable map from $(X,\mathscr A)$ to itself. As far as I am not mistaken, the space of solutions of $(1)$ is a monoid as it is closed under the composition of maps.


Also, if $\xi$ is a bijection which solves $(1)$ then $\xi^{-1}$ solves it as well:
$$
\xi^{-1}\_\*(\mathsf Q)(A) = \mathsf Q(\xi(A)) = \mathsf Q(\xi^{-1}(\xi(A))) = \mathsf Q(A).
$$


Hence, bijective solutions of $(1)$ form a group - which may seem to be thought of a group of "symmetries" of $\mathsf Q$. For example, the standard normal distribution over reals $\mathsf Q = \mathscr N(0,1)$ admits at least two representations $\xi(\omega) = \omega$ and $\xi(\omega) = -\omega$. As well as any Haar measure over a group admits representation via $\xi(\omega) = \alpha \omega$ where $\alpha$ is any element of the group.


I've asked this question on [MSE](https://math.stackexchange.com/questions/186259/uniqueness-of-the-random-variable-from-its-distribution), but I have not received any answers.




---


**Edited:** To clarify (as requested), my question is exactly as follows: are such groups of symmetries of measures studied somewhere in the literature - may be, providing some interesting results for measures exhibiting such symmetries. I have studied the Lie groups of ODE/PDE symmetries, and I wonder if there is anything similar known for measures.



Symmetries of probability distributions
This is a fascinating topic. One impressive systematic study of symmetries is in the book by Olav Kallenberg (2005)


* [Probabilistic Symmetries and Invariance Principles](https://rads.stackoverflow.com/amzn/click/com/0387251154)


In there, though, the measurable space has to have some structure to get the most out of the results.


I don't know of any systematic applications of Lie groups to probability theory. However, there are here and there some interesting results. For instance, this book contains a study of measures that are invariant under O(n).


* [Fang, Kotz and Ng (1990). Symmetric multivariate and related distributions](https://rads.stackoverflow.com/amzn/click/com/0412314304)


There is also plenty of results and applications of discrete symmetries (among others) in here:


* [Diaconis (1988). Group representations in probability and statistics](https://projecteuclid.org/euclid.lnms/1215467407)


Maybe one should ask a community wiki question where everyone tries to list the results they know. That would be a very interesting list!


**Edit**: I recently came across this book that is a quite relevant reference for studying symmetries of probability measures:


* [Wijsman (1990). Invariant measures on groups and their use in statistics](https://projecteuclid.org/euclid.lnms/1215540653)


It has an extensive discussion on Lie groups ans Lie algebras.


**Edit 2**: Another book with an extensive discussion on Lie groups in Probability and Statistics!


* [Chikuse (2003). Statistics on Special Manifolds](https://rads.stackoverflow.com/amzn/click/com/0387001603)


