Let $(\Omega,\mathcal{F}, P)$ a probability space with $P: \mathcal{F} \to [0,1] $. In my understanding, measuring the probability of an event (any kind event) is equivalent to measure the "size" of a set in $\mathcal F$. Also, in my understanding, thanks to the Radon-Nikodym theorem we can convert the probability measure $P$ into the Lebesgue measure over $\mathbb R$. In other works, for any set $A\in \mathcal F$ we have $$ P(A)= \int_{A}dP=\int_{X(A)}f_X(x)dx $$ By keeping in mind the conversion "probability measure" $\to$ "Lebesgue measure on the real-line", from the above equality one can easily grasp that $X(A)$ is a subset of $\mathbb R$. Therefore, I find more natural to define $X : \mathcal F \to C \subset 2^{\mathbb R}$, where $C$ is appropriately chosen, rather than $X : \Omega \to \mathbb R$. That way, it is easier to grasp that $X$ maps events into intervals of the real-line. Does it make sense?
Furthermore, why $f_X$ is a distribution? My intuition would suggest that $f_X$ is a distribution because in reality (considering the standard definition of random variable) it should be written $P(A)=\int_{X(A)}f_X(X(\omega))dX(\omega)$, but I am not sure if it this is actually the reason. If so, should the "x-axis" of the plot of $f_X(x)$ represent $X(\omega)$?
