Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

Question

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to:

Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a profinite group $G$, $\pi_i : G \to G / N_i = G_i$ the natural projections, and $U$ an open set. Then $\mu(U) = \lim |\pi_i(U)| / |G_i|$, where $\mu$ is the Haar measure, and $||$ is the counting measure on the quotient.

Is this correct? What exactly does "lim" mean there - limit as a net in the real numbers?

I think it is more or less clear that such a function is bi-invariant, and has $\mu(G) = 1$. I'm less sure about how to prove that this is countable additive - it's clear for cosets of open subgroups, but I don't see how to jump from those to all open sets. I guess one would also have to prove some kind of regularity, or maybe those follow from measure theoretic machinery - I don't remember these details of measure theory too well.

Anyhow, I would appreciate a reference or an explanation!

Answer 1

Sorry for the source, but Wikipedia seems to give what you want:

https://en.wikipedia.org/wiki/Haar_measure#A_construction_using_compact_subsets

Take A to be your profinite group and the directed set of U's to be your $N_i$'s. The delicacy of the countable additivity is summarized pretty well, I think. Weil fleshes out the details, but his exposition seems to be in "L'intégration dans les groupes topologiques et ses applications" and Google didn't show me an easy way to look in that book.

Answer 2

Will add more details later, but this should be a consequence of the uniqueness in Kolmogorov's extension theorem, which will show that the limit of the counting measures is invariant.