Is it true that every subgroup of finite nonzero 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.

2$\begingroup$ I think that it's classical that if $X$ is a measurable subset of nonzero measure then $XX$ has nonempty 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). $\endgroup$– YCorJan 31, 2016 at 13:54

$\begingroup$ By $XX$ I mean the set of $xy$ when $x,y\in X$, which of course is the same as $X+X$ if $X$ is symmetric. $\endgroup$– YCorJan 31, 2016 at 14:35
1 Answer
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 $G$.

$\begingroup$ How do you show that $\nu$ is a Haar measure? namely, that it takes finite values on finite subsets? $\endgroup$– YCorSep 10, 2016 at 6:47

$\begingroup$ 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. $\endgroup$ Sep 10, 2016 at 7:02

$\begingroup$ oops sorry I did a typo, I meant "how do you show that $\nu$ takes finite values on compact subsets"? $\endgroup$– YCorSep 10, 2016 at 7:14

$\begingroup$ 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. $\endgroup$ Sep 10, 2016 at 9:06

$\begingroup$ Ah ok sure. This is properness of the quotient morphism when the kernel is compact. $\endgroup$– YCorSep 10, 2016 at 9:16