Visualizing a probability measures through a probability density functions I found a previous question with a very nice answer, but still there is something that is not completely clear to me.
We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let $\Delta (X)$ denote the set of all probability measures over $X$. 
Can we say that a specific probability density function is a way to graphically represent an element $p \in \Delta (X)$? 
Thank you for your time.
 A: 
"To measure density one must have a way to measure volume as density is the proportion of mass per unit of volume."

Therefore we need a special $\nu\in\Delta(X)$ which is going to be the "volume" measure.
I will do the explanation with the most common example of $X=\mathbb{R}$ (or $\mathbb{R}^n$). On $\mathbb{R}$ (or $\mathbb{R}^n$) we have the Lebesgue measure $\nu$ which is the standard measure of length.
For an element $p\in\Delta(\mathbb{R})$ to have a density with respect to $\nu$ we need that $p$ is absolutely continuous with respect to $\nu$. This means that if $E$ is a measurable set and $\nu(E)=0$ then $p(E)=0$ too. It happens that not all $p\in\Delta(\mathbb{R})$ have a density.
For those $p\in\Delta(\mathbb{R})$ that are absolutely continuous with respect to $\nu$ there is a function $f$ (by Radon-Nikodym's theorem) such that $p(E)=\int_E f(x)d\nu(x)$. That function is what is called the density of $p$ (with respect to $\nu$).

The connection with visualization is that in $\mathbb{R}^n$ you can graph the function $f$. But there is another part of the story.
Suppose that $X$ is not necessarily $\mathbb{R}^n$. But we have a random variable $Y$ (a measurable function) on $X$ that takes values on $\mathbb{R}^n$, $Y:X\to\mathbb{R}^n$. Then one can use $Y$ to export measures on $X$ to measures on $\mathbb{R}^n$, and use those measures to study $Y$. If $E\subset\mathbb{R}^n$ is measurable we can put $P(E):=p(Y^{-1}(E))$. This gives us a measure $P$ on $\mathbb{R}^n$. Since on $\mathbb{R}^n$ we have the special measure, the Lebesgue measure $\nu$ then we can consider (if it exists) the density of $P$ with respect to $\nu$.
