Probability measure on the Rationals. density function, and integration? 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: 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.
