Reference request for locally compact groups proof

Question

Where can I find a proof of the following:

"Every locally compact group is a directed union of $\sigma$-compact open subgroups"

This is claimed in Greenleaf's book but I haven't been able to find a proof.

Thanks!

Answer 1

I believe the following is a proof. Let $G$ be a locally compact group and $\mathcal{K}$ be the collection of all compact subsets of $G$ with nonempty interior (directed by inclusion). It's easy to see that the group $\langle K\rangle$ generated by $K$ is open for every $K\in\mathcal{K}$, and since $G$ is locally compact every element is in $K$ for some $K$ (and hence in $\langle K\rangle$). Thus $G$ is the directed union of these $\langle K\rangle$ (as $K$ runs over $\mathcal{K}$).

I hope I understood the question properly.