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.