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](https://en.wikipedia.org/wiki/Haar_measure) $S\_\omega$.
> 
> 
> 


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 $S\_\omega \setminus M$?



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


