Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.

• I think that it's classical that if $X$ is a measurable subset of nonzero measure then $X-X$ has non-empty interior? If so it's clear that any measurable subgroup of positive measure has to be open (and hence compact open if it has finite measure).
– YCor
Jan 31 '16 at 13:54
• By $X-X$ I mean the set of $x-y$ when $x,y\in X$, which of course is the same as $X+X$ if $X$ is symmetric.
– YCor
Jan 31 '16 at 14:35

Let $G$ be an LCA group, $\mu$ be a Haar measure on $G$ and $H$ be a (closed) subgroup of $G$ with $0<\mu(H)<\infty$. First, by restricting $\mu$ to $H$, we obtain a finite Haar measure on $H$. This means that $H$ is compact. Further, by projecting $\mu$ to $G/H$ via the quotient morphism $G\to G/H$, we obtain a Haar measure $\nu$ on $G/H$, which is positive on singletons. This means that $G/H$ is discrete, hence $H$ is open in $G$.

• How do you show that $\nu$ is a Haar measure? namely, that it takes finite values on finite subsets?
– YCor
Sep 10 '16 at 6:47
• Formally, $\nu$ is defined by $\nu(A)=\mu(p^{-1}(A))$ for $A\subseteq G/H$ Borel, where $p\colon G\to G/H$ is the quotient morphism. Then $\nu(\text{singleton in$G/H$})=\mu(\text{$H-$coset in$G$})=\mu(H)$, which is positive and finite by the assumptions. Sep 10 '16 at 7:02
• oops sorry I did a typo, I meant "how do you show that $\nu$ takes finite values on compact subsets"?
– YCor
Sep 10 '16 at 7:14
• This follows from compactness of $H$ (which is established in the beginning). If $K\subseteq G/H$ is compact then $p^{-1}(K)$ is also compact. Indeed, first, there is a compact set $C\subseteq G$ with $p(C)=K$. Now $p^{-1}(K)=CH$ is a product of compact sets and hence is compact. Thus, $\nu(K)=\mu(CH)$ is finite. Sep 10 '16 at 9:06
• Ah ok sure. This is properness of the quotient morphism when the kernel is compact.
– YCor
Sep 10 '16 at 9:16