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29 March 2024
 
  » arxiv » 1607.0049

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The Steinhaus-Weil property: its converse, Solecki amenability and subcontinuity
N. H. Bingham ; A. J. Ostaszewski ;
Date 30 Jun 2016
AbstractThe Steinhaus-Weil theorem that concerns us here is the ’interior points’ property -- that in a topological group a non-negligible set S has the identity as an interior point of $SS^{-1}$. There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar measure. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. In Part I (Propositions 1-9, Theorems 1-3) we develop a number of relatives of the Simmons-Mospan theorem, drawing also on Solecki’s amenability at 1 (and using Fuller’s notion of subcontinuity). In Part II (Theorems 4, 5) we link this with topologies of Weil type.
Source arXiv, 1607.0049
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