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?


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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} $.