How common are probability distributions with a finite variance? It's always very surprising to learn that some of the entities one has been assiduously studying actually represent negligibly tiny minorities (e.g. continuous functions vis-à-vis all functions)... * * * Now, the Central Limit Theorem, for one, holds only for probability distributions with a finite variance. How common are such distributions in the space of all probability distributions? * * * More formally, let $U$ be the set of all probability distributions on $\mathbb{R}$ (say), and $F \subset U$ be the subset of those probability distributions that have a finite variance. I expect that, of the cardinalities $|F|$ and $|U\setminus F|$, one will be a strictly larger infinity than the other, but I have no intuition as to which. * * * (I guess this is a "meta-measure theory" question.)

I presume that you speak about distributions on the Borel $\sigma$-algebra $\mathcal B(\mathbb R)$. In terms of cardinality, there are quite few of them, namely, $\mathfrak{c} = |\mathbb{R}|$. Indeed, each probability distribution $\mathcal B(\mathbb R)$ is uniquely determined by its values on the intervals $(-\infty,q)$ with $q\in\mathbb{Q}$. Therefore, the number of probability measures does not exceed the number of sequences of real numbers, which is $\mathfrak{c}$. Since, obviously, there are both $\mathfrak{c}$ distributions with finite and infinite variance, so your guess is wrong.

There are other ways to compare sets, e.g. in terms of the Baire categories. If we consider the weak convergence topology, then both finite variance and infinite variance sets are easily shown to be second category, so another draw here.

Overall, I see no reason to believe that there are fewer finite variance (or finite expectation or finite exponential moment) distributions than others.