I'm just thinking about it. For example, a bijection between $\mathbb{Z^*_+}$ and $\mathbb{Q}$ implies that the probability of a random real number being rational or positive integer is the same (in this case, 0)? And if it's true, how to prove it?
Another question: Suppose (for example) $\mathbb{R^{1/2}}$ is a generic subset of $\mathbb{R}$ such that the probability of an element of $\mathbb{R}$ being also an element of $\mathbb{R^{1/2}}$ is $1/2$. There are studies of probabilistic cartesian products between those generic sets?
Consider the normal distribution. You can define the bijection $f(x)=e^x$ from $\Bbb R$ to $(0,\infty)$. The probability of the former is $1$ and of the latter, $1/2$.