Given a random variable $X$, we know that its distribution function $F_X$ exists. But its density function $f_X$ may not exist. I am thinking about conditions that allow $f_X$ to exist.
From the definition, a R.V. $X$ has a pdf $f_X$ if and only if $F_X$ can be expressed as $$F_X(a) = \int_{-\infty}^a f_X(x)dx ~ ~ ~ \forall a \in \mathbb{R}.$$
From measure theory, a monotone increasing function $F_X$ can be written as an indefinite integral like above, if and only if $F_X$ is absolutely continuous. (Or equivalently, the Lebesgue-Stieltjes measure $\mu_F$ generated from $F_X$ is absolutely continuous w.r.t the Lebesgue measure $\lambda$.)
Can I understand that the condition that $f_X$ exists is that its corresponding distribution function $F_X$ must be absolutely continuous?
It's well known Radon–Nikodym theorem and can be found in Halmos P.R. "Measure Theory" 128p, or here:
Given $(X, S, \mu)$ $\sigma$-finite measure space and $\sigma$-finite signed measure $\nu$ absolute continuous with respect to $\mu$, then exists measurable $f$, such that $$\nu(E)=\int\limits_{E}fd\mu$$ for every measurable set $E$.