uniform probability distribution on the reals

Question

I heard that you cannot create a uniform probability distribution on the reals because it breaks the additivity axiom where the individual probabilities of a countable number of disjoint subsets of your space should equal the probability of their union. So how would you mathematically describe an idea such as selecting a random number from the reals? Or ideas similar to this. Is there some other kind of math that you use, some other distribution or concept?

Answer 1

(1) How about: uniform distribution on an interval $[a,b]$?

(2) More generally, fix a probability distribution $\phi$ on $\mathbb R$, then use $\phi$ to specify your random numbers. For example, use a normal distribution.

(3) Do not use a probability measure, instead use Lebesgue measure.

More philosophical answer: why would you want to choose a random real number? Use those reasons to specify what properties you want.

Answer 2

While it is not a "distribution" in the strict sense, there is the notion of an improper prior. This corresponds not to a specific random selection procedure, but rather to a general state of ignorance about the details of said selection procedure.