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} $.