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I've been reading the classical Furstenberg paper "The structure of distal flows", where the author claims he is working with an arbitrary locally compact group $T$. Nevertheless, the proof of Lemma $5.1$ actually requires that $T$ is $\sigma$-compact. I am not aware of any proof about locally compact groups acting distally on compact metric spaces being $\sigma$-compact.

The question is: is there such a proof, or does this mean his theorems about quasi-isometric flow have a lesser degree of generality?

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If a topological group is locally compact and connected, then it is $\sigma$-compact. I did not read the paper but the word "flow" suggests that everything there is connected. – Sergei Ivanov Sep 23 '12 at 15:43

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