I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are distributed. Is it correct to formalize the problem in the following way?
Let $m$ be a probability measure on $\mathbb{R}^n$, and let $X$ be a $n\times n$ matrix whose entries are i.i.d. Gaussian random variables. $X$ can be seen as a random variable whose law is a probability measure on $\operatorname{Mat}(n,n)$, so by taking the product we have a probability measure on $\operatorname{Mat}(n,n)\times\mathbb{R}^n$.
If we pushforward the probability on this space using the matrix multiplication $\operatorname{Mat}(n,n)\times\mathbb{R}^n\to\mathbb{R}^n$, sending $(M,x)\to Mx$, we get a new measure that we can denote $m_X$ on $\mathbb{R}^n$.
My question is, what is known about this new measure? It seems to me a very natural object to study, so I would expect some theory to exist about it, but I couldn't find it.
All I would say is that if $m$ is absolutely continuous wrt to the Lebesgue measure, so is $m_X$, since $X$ is almost surely nondegenerate. Also I think that $m_X$ might be $O(n)$-invariant since I imagine this measure to me an average over all possible matrices. But for example what can we say about the volume of a ball of radius $R$ and centered in the origin $m_X(B(0,R))$?
