Haar-open sets: a right way of generalizing the Steinhaus sum theorem to non-locally compact groups

2018/05/19 by Banakh, Taras
#28C10 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1805.07515

Abstract

Let X be the countable product of Abelian locally compact Polish groups and A,B⊂ X be two Borel sets, which are not Haar-null in X. We prove that the sum-set A+B:=\a+b:a∈ A, b∈ B\ is Haar-open in the sense that for any non-empty compact subset K⊂ X and point p∈ K there exists a point x∈ X such that the set K∩(A+B+x) is a neighborhood of p in K. This is a generalization of the classical Steinhaus Theorem (1920) to non-locally compact groups. We do not know if this generalization holds for Banach spaces.

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