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 $S_\omega \setminus M$?