RV without pdf but $P(X=x)=0$ for all $x\in \mathbb{R}$ Find a probability space $(\Omega, \mathcal{F},P)$ and a random variable $X:\Omega \rightarrow \mathbb{R}$ such that $P(X=x)=0$ for all $x\in \mathbb{R}$, while $X$ does not attain a probability density function.
Extra: does such an RV exit in every probability space?
Thanks a lot in advance!  
(Some thoughts: I started with the Dirac measure $\delta_0$ (which does not attain a density function) on $(\mathbb{R},\mathcal{B})$, but an RV that has the cdf $F$ attained by $\delta_0$ is the constant RV $X(\omega)=\min\{x: F(x)\geq \omega\}=0$, so $P(X=0)=1$, while $P(X=x)=0,~x\neq 0$).

What you are looking for is actually an example for a random variable (or a distribution) which is continuous, but not absolutely continuous. One example for such a distributuon is the so-called Cantor distribution.
Note that, by the Radon-Nikodym theorem, a distribution $\mu$ is absolutely continous (with respect to the Lebesgue measure) if, and only if,
$$\mu(A)=0$$
for any (measurable) set $A$ with Lebesgue measure zero. If we assume that $\mu(\{x\})=0$ for all $x$, this yields $\mu(A) = 0$ for all countable sets $A$, but not necessarily $\mu(A) = 0$ for all Lebesgue-null sets.
