"Haar-like" measure on $S_\omega$

Question

Let $S_\omega$ be the collection of bijections $f:\omega\to \omega$. Endow $\omega$ with the discrete topology and let $S_\omega$ be endowed with the subspace topology of $\omega^\omega$, where $\omega^\omega$ carries the product topology.

EDIT. The following statement of mine from the original post is false:

False statement: $(S_\omega, \circ)$ is a locally compact group and so there is a Haar measure on $S_\omega$.

But: I would nevertheless like to know whether there is some "Haar-like" measure on $S_\omega$. Is there a constructive description of such a measure?

If yes: Let $M$ be the set of "finitely bounded permutations of $\omega$, that is, $$M=\{\pi \in S_\omega: \exists K\in\omega(\forall n\in \omega(|\pi(n)-n| < K))\}.$$ What is the Haar measure of $M$, and of $S_\omega \setminus M$?

Answer 1

Whenever $G$ is a non locally compact Polish group, there does not exist any nonzero $\sigma$-finite Borel measure $\mu$ on $G$ such that all left-translates of $\mu$ are absolutely continuous with respect to $\mu$. This is due to Weil; there is a nice proof in a 1946 paper of Oxtoby.

Answer 2

You'll have to restrict your class of observables. What you can do for example is look at a uniformly chosen permutation of $N$ elements and only keep track of its cycle decomposition. You'll find that most points belong to cycles of length about $\alpha N$ for some $\alpha \in (0,1)$. If you then only keep track of cycle lengths, divide their length by $N$, and let $N \to \infty$, you do get a limiting distribution on the space of unordered partitions of $[0,1]$ called the Poisson-Dirichlet distribution. As usual, Terry has a nice write-up on his blog.