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, 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.
Best Answer
This is a fascinating topic. One impressive systematic study of symmetries is in the book by Olav Kallenberg (2005)
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).
There is also plenty of results and applications of discrete symmetries (among others) in here:
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:
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!