When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?

Question

Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed.

A Remark in Folland's, A Course in Abstract Harmonic Analysis, Section $2.3$, claims the restriction of a left Haar measure $\lambda$ on $G$ results in a left Haar measure on $H$.

I haven't been able to come up with a proof of this fact. For a general subgroup $H$ of $G$ this remark fails to be true, so how do the properties assigned to $H$ make this true?

Answer 1

This will work if the subgroup is open, so that it has positive measure. (Open subgroups are automatically also closed.) What problems have you had in the proof? Surely it is easy to show that this restriction is left-invariant?

added
There would be a converse, too: a (measurable) subgroup with positive measure must be open.