Probability that an element belongs to an infinite subset of a set

Question

$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an element" has a sense, and it is the smallest positive integer $n$ such that $f^n = e$, where $f$ is one of those bijections and $e$ is the identity of the group (the identity function).

I was interested in the subset of all the elements of the group that have finite period. My question is: if I randomly choose an element of $\Sym(\mathbb{N})$, is there a way to know... if it's more likely to get an element of infinite period, or an element of finite period? The problem is that, according to the results I got, both $\Sym(\mathbb{N})$ and its subset I'm interested in are infinite sets that have the cardinality of the continuum.

Am I unawarely asking a stupid/impossible question, or are there any mathematical tools to know what that probability is?

Answer 1

Maybe the hardest part of answering this question is deciding whether there's some natural meaning of the concept of randomly choosing an element of this set.

The set in question is uncountably infinite.

Under the most usual conventions of probability, one wants a probability measure on some sigma-algebra of subsets of the space in question. Which sigma-algebra should be used in this case? Is there some natural answer to that question. Should we regard the set of all permutations that map $3$ to $8$ (and others like that) as a sub-basic open set, and the set of all unions of finite intersections of such sets as open, and then look at Borel sets in that topological space?

After that, which probability measure on the sigma-algebra should we assign?

What, for example, is the probability that $3$ is mapped to $8$? If $\tau$ is a randomly chosen permutation, then certainly $$ \sum_{n\in\mathbb N} \Pr(\tau(3)=n) = 1. $$ But the probabilities $\Pr(\tau(3)=n)$ for $n\in\mathbb N$ would differ from each other.

Maybe there are lots of interesting probability distributions on this set, just as on the sigma-algebra of Borel subsets of $\mathbb R$. We all learn about the normal distribution, the exponential distribution, etc., etc., etc.

So there's still a lot of work to do before you have a well-defined mathematical question.