Why isn't there a uniform probability distribution over the positive real numbers? Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what probability distributions can exist over the positive real numbers (it seems that this would be quite limited, given that such a simple distribution is impossible)?

For every probability density $f$, $\liminf_{x\to \pm \infty} f(x) = 0$ must hold.
It is clear that for a uniform distribution, the density has to be constant on the considered interval.
Combining both requirements, only $g(x) = 0$ remains as a choice for a uniform density on $(-\infty, \infty)$. But $\int_{-\infty}^\infty g(x)dx = 0 \neq 1$, that is $g$ is no proper probability density.
