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permutations – “Haar-affection” touchstone on $S_omega$ retort

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permutations – “Haar-affection” touchstone on $S_omega$

Let $S_omega$ breathe the gathering of bijections $f:omegato omega$. Endow $omega$ with the discrete topology and let $S_omega$ breathe endowed with the subspace topology of $omega^omega$, the place $omega^omega$ carries the product topology.

EDIT. The following assertion of mine from the unique put up is pretend:

pretend assertion: $(S_omega, circ)$ is a domestically compact group and so there’s a Haar touchstone on $S_omega$.

But: I might nonetheless affection to know whether or not there’s some “Haar-affection” touchstone on $S_omega$. Is there a constructive description of such a touchstone?

If sure: Let $M$ breathe the clique of “finitely bounded permutations of $omega$, that is, $$M= < K)).$$ What is the Haar touchstone of $M$, and of $S_omega setminus M$?

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