# Thread: Closed subgroups of R

1. ## Closed subgroups of R

Find all the closed subgroups of (R,+)? And if H is any proper subgroup of (R,+) then prove that m(H)=0, where m = lebesgue measure.

2. Originally Posted by Chandru1
Find all the closed subgroups of (R,+)? And if H is any proper subgroup of (R,+) then prove that m(H)=0, where m = lebesgue measure.
Closed in what sense? Closed when considering $\mathbb{R}$ as a topological group?

3. ## yes

Yes.

4. Originally Posted by Chandru1
Find all the closed subgroups of (R,+)? And if H is any proper subgroup of (R,+) then prove that m(H)=0, where m = lebesgue measure.

Read here A course in p-adic analysis - Google Books . You need exactly the corollary at the top of page 23, and from it it follows at once that the Lebesgue measure of any closed sbgp. is zero.

Tonio