# Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff groups we should have a similar theorem, which would tell us $\{g^{-1}h:g,h\in E\}$ for any subset with positive measure should contain an open neighborhood of the identity.

It seems to me this theorem should be able to tell us a lot about the structure on a topological group, but I cannot really find one example.

Can someone point to some nice applications of Steinhaus in topological groups?

Thanks!

#### Solutions Collecting From Web of "Is Steinhaus theorem ever used in topological groups?"

I’m not sure, if this sufficies for an answer..anyhow, I think it’s a neat corollary. I believe this is due to Mackey:

Suppose you have two locally compact, Hausdorff, second countable groups $G, H$ and a measurable homomorphism $\phi \colon G \to H$. Then $\phi$ is actually continuous.

To prove this, start with open neighborhoods $U,V$ of $e_{H}$ such that $VV^{-1} \subset U$. Using the second countability of $G, H$, we get a countable set $\{ g_{n} \}_{n \in \mathbb{N}}$ such that $G = \cup_{n \in \mathbb{N}} g_{n} \phi^{-1}(V)$. Hence there is a $n_{0}$ with $m_{G}(g_{n_{0}}\phi^{-1}(V)) > 0$ and then $m_{G}(\phi^{-1}(V)) > 0$ by left invariance. Now as

$\phi^{-1}(V) \phi^{-1}(V)^{-1} \subset \phi^{-1}(VV^{-1}) \subset \phi^{-1}(U)$

we see by Steinhaus that $U$ contains an open neighborhood of $e_{G}$.