Galois groups are nice compact Hausdorff groups, and therefore possess a bounded Haar measure, unique if we insist that the total volume be $1$. What is the Haar measure on the absolute Galois group of $\mathbb{Q}$ (or of any other number field)? Is there any arithmetic meaning for a subgroup being "big" or "small" with respect to this measure?
Two remarks:
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1. In the book Field Arithmetic by M. Fried and M. Jarden there is a small treatment of Haar measures on Galois groups in $\S 18.5$, but I did not find much a part from general statements which, in case the base field is $\mathbb{Q}$, do not look that much sexy (to me).
2. Following Tate's thesis, it seems natural to put on the abelianization $$ G_\mathbb{Q}^\text{ab}=\mathrm{Gal}(\mathbb{Q}^\text{ab}/\mathbb{Q})=G_\mathbb{Q}/[G_\mathbb{Q},G_\mathbb{Q}] $$ the quotient measure coming from first putting on the ideles group $\mathbb{A}_\mathbb{Q}^\times$ the "product Haar measure" coming from each local factor – this is what Tate does, indeed (observe that the normalization is not trivial, here, since the ideles are only locally compact); and, then, identifying the group $G_\mathbb{Q}^\text{ab}$ with a quotient of $\mathbb{A}_\mathbb{Q}^\times$ via global Class Field Theory. But then
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i) Does the Haar measure on $\mathbb{A}_\mathbb{Q}^\times$ really descend to a Haar measure on its quotient $G_\mathbb{Q}^\text{ab}$?
ii) if i) holds, we get a Haar measure on $G_\mathbb{Q}^\text{ab}$, with respect to which it has some volume $vol_{ideles}$. Similarly, we can hope that the Haar measure on the absolute Galois group induces a Haar measure on its abelianization, the latter being a quotient of the former: this will give $G_\mathbb{Q}^\text{ab}$ another volume $vol_{galois}$. Are the values $vol_{galois}$ and $vol_{ideles}$ related (and interesting)?
