(1) Let $(\Omega,\mathcal{F},P)$ be a probability measure space and $X:\Omega \rightarrow \mathbb{R}$ a random variable. Let $P_X,~F_X$ denote the probability measure, pdf induced by $X$, respectively. Under which conditions on X (or $P_X$, $F_X$) does X admit a density function. i.e. an integrable function $f$ such that $\displaystyle P(X\in B)=P_X(B)=\int_Bf(x)dx,~B\in \mathcal{B}$?
PS. I am aware of the Radon - Nikodym theorem, introducing conditions under which a (probability) measure generally admits a density function, but do they change when dealing with a probability measure induced $P_X$ by a rv $X$ (and how)?
(2) Let $F$ be a probability measure on $(\mathbb{R},\mathcal{B})$. Under which conditions, if any, does there exist a random variable X from some probability space $(\Omega,\mathcal{F},P)$, such that $F=F_X$, i.e. $F$ is the pdf induced by X?
Thanks a lot in advance for the help!