I'm reading some texts about lattices in topological groups, and Wikipedia has a discussion under "Definition" in the link https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup).
I understand the two sub-paragraphs, but fail to understand:
In the case of discrete subgroups this invariant measure coincides locally with the Haar measure and hence a discrete subgroup in a locally compact group $G$ being a lattice is equivalent to it having a fundamental domain (for the action on $G$ by left-translations) of finite volume for the Haar measure.
I've been struggling a lot with this and couldn't understand it. It is mentioned (without proof) in many books, so it's probably correct.
Why does this measure coincide locally with the Haar measure? As far as I can see, there's no reason for this measure to be inner+outer regular, which I think is required.
Why does it imply the existence of a fundamental domain?
Any comment would be appreciated