How do you define a p.d.f and c.d.f for a random variable that realises real values from real line with no bias?

Question

Basically assume that it is given to you that random variable X outputs real values in R and each value is equilikely. Then how could it's p.d.f and c.d.f be defined. Also once that is done for R, then can it we done for set of all integers as well i.e. Z

Answer 1

A positive, non-zero and translation-invariant measure on the Borel subsets of $\Bbb R$ cannot be finite, so there is no translation-invariant Borel probability $\Bbb R$.

Answer 2

It would be nice if we could do this, but it's impossible.

For $\Bbb Z$ it's easy to see: if $p$ is the probability of choosing any integer $n$, then (by countable additivity) the sum of the probabilities over all integers $n$ would either be $0$ (if $p=0$) or $\infty$ (if $p>0$), both of which contradict the fact that the total probability must equal $1$.

For $\Bbb R$ it's a little harder to see, since there are uncountably many possible values. However, presumably such a random variable would give equal probability $p$ to each of the intervals ..., $[-1,0)$, $[0,1)$, $[1,2)$, ...; and then the same contradiction can be reached as above.