Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space.

Usually, we have a probability space $(\Omega, \mathcal{F}, P)$, a random variable $X:\Omega\to\mathbb{R}$ and the PDF of $X$, $f_X:\mathbb{R}\to[0, \infty)$. The whole point of the PDF is that we can integrate it over any borel set $A$ (contained in the codomain of $X$) and get the probability $ P[X\in A] = \int_A f_X\ dx$.

What about the case when we have that $\Omega = \mathbb{R}$ and $\mathcal{F}$ is the Borel sigma algebra over $\mathbb{R}$? Looks like it's easier to define the PDF to be a function $f:\Omega\to[0,\infty)$ such that $P[A] = \int_A f\ dx$. So we still can use this PDF to get the probability of events via integration over borel sets. The idea in the book (mentioned on the other post) is that $\Omega$ can be a manifold, so we integrate the PDF over a subset contained in this manifold to get the probability of that subset. If $X:\Omega\to\mathbb{R}$ is a random variable (not necessarily the map $x\to x$), we could talk about $f_X$, the PDF of $X$, such as described before. But in this sample space, we can use the other PDF $f$. So, instead calculating $\int_Af_X\ dx = P[X\in A]$, we can calculate $\int_{\{X\in A\}}f\ dx = P[X\in A]$. Note that the first integral is over the codomain of $X$, but the second is over the sample space $\Omega$.

This article in particular is an example of what I'm saying. Instead defining real random variables $X_{ij}$ as the entries of a random matrix, it's possible to see the sample space as $\mathbb{R}^{n^2}$ and define a probability distribution on $\mathbb{R}^{n^2}$ to work with.

I want to know if there is some technical detail I should be aware of and how common is this. Thank you.