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A famous Steinhaus theorem says that if measurable subsets $A,B$ of a locally compact topological group $G$ have positive Haar measure, then the difference $AA^{-1}$ is a neighborhood of the unit and …

I have just realized that this my question has a simple negative answer: Denote by $\mathbb R_+=[0,\infty)$ the half-line. Observe that the countable product of lines $G=\mathbb R^\omega$ is an Abelia …

It seems that the answer to this problem is affirmative for compact groups. Indeed, if the union $E$ of $H$-cosets is measurable in $G$, then for every $\epsilon>0$ we can find two compact subsets $A\ …

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the s …

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in …

In the Frechet space $X:=\mathbb R^\omega$ consider the dense linear subspace $$L_0:=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:|\{n\in\omega:x_n\ne0\}|<\omega\}.$$ Fix a countable base $\{V_n\}_{n\in\o …

The answer is NO because the Euclidean and the discrete topologies are the unique locally compact group topologies on $\mathbb R$, which are stronger that the Euclidean topology of the real line. Th …