Uniform measure on the rationals between 0 and 1 I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the same probability. 
Of course from the fact, that there are infinitely many atoms, it follows, that the measure has to be zero for every rational, which in turn yields a zero probability for any event, even the whole space, which is bad.
So is something wrong about my idea of uniformity?
What kind of measures can be defined on $X$?
How could one design a "completely random drawing of rationals"?
(Do we appreciate what we have with the reals, when we talk about uniformly distributed random number?)

[Comment promoted to answer at request of OP] 
No countably infinite set supports a uniform measure (other than the measure where every set has measure zero, and the one where each element has the same non-zero measure and the whole set has infinite measure). Drawing a rational uniformly at random is as impossible as drawing an integer uniformly at random.
