Nornally, we define a probability space on the continuum, the real numbers.
For simplicity, lets say we have a "flat distribution" on $[0,1]\subset \mathbb R$. Then the measure of the rationals $[0,1]\subset \mathbb Q$ is zero.
But what if we instead define the measure only on the rationals? That is, for any set containing only irrationals, that sets measure is zero. And the measure of $ [0,1]\cap \mathbb Q$ is 1.
Can we define, without problems, the same density function on [0,1] as we would if we were using the real numbers?
Can we define the cumulative distribution function in the same way, without problems?
Does this in any other way generate pathologies? Or is it legitimate?
A probability measure $\mathsf P$ on $\langle\mathbb R^n,\mathcal B^n\rangle$ will only have a probability density function if it is absolutely continuous wrt the Lebesgue measure.
That is, if: $$\lambda(B)=0\implies\mathsf P(B)=0$$ for any measurable set $B$, where $\lambda$ denotes the Lebesgue measure.
Note that this is a necessary condition simply because: $$\lambda(B)=0\implies\mathsf P(B)= \int_Bf(x)\lambda(dx)=0$$ for any probability density $f$ of $\mathsf P$.
We have $\lambda([0,1]\cap\mathbb Q)=0$ so if $\mathsf P([0,1]\cap\mathbb Q)=1$ then no density wrt the Lebesgue measure exists.
In that situation a so-called probability mass function comes in.
On $[0,1]\cap\mathbb Q$ we have the counting measure sending every set $A\subseteq[0,1]\cap\mathbb Q$ to its cardinality in $\{0,1,2,\dots\}\cup\{\infty\}$. In this situation the probability measure is absolutely continuous wrt to the counting measure and also has a density wrt that measure.
This is a function $p:[0,1]\cap\mathbb Q\to[0,\infty)$ such that $P(A)=\sum_{a\in A}p(a)$ for every $A\subseteq[0,1]\cap\mathbb Q$.
It is more common to define it as a function $\mathbb R\to[0,\infty)$ such that can only take positive values on elements in $[0,1]\cap\mathbb Q$.
So $p$ is somehow a density (wrt counting measure) but is not a density wrt the Lebesgue measure. It never gets the label PDF but it gets the label PMF.