The generalization to G-flows should be easy once you have this.

Point 3, The pdf file you found is not a particularly good source, even though everything in it seems to be correct. I will think about a better one. The connection between homomorphisms and ultrafilters is this: suppose for simplicity that the group is Z. if you have an ultrafilter U on Z and a bounded function f on Z, you can form the U-limit of f: just start with a finite interval such that all values of f belong to it, divide it in half, and choose the half such that U-majority of the values of f belong to that half. Then cut that half in half again according to the same rule, and keep doing it until you get a single point –it is the unique real r such that the f-preimage of any neighborhood of x is in U. the map f to U-lim f is a continuous homomorphism. On the other hand, if you see a continuous homomorphism h, just consider the collection of those sets such that h of their characteristic function gives 1. Check that this is an ultrafilter.

I will get to the other points later

]]>I skimmed it over and it looks helpful, so I’ll take a look at it a little closer and see if I still have questions.

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