The Steinhaus-Weil Property: I. Subcontinuity and Amenability

The Steinhaus-Weil theorem that concerns us here is the simple, or classical, `interior-points' property -- that in a Polish topological group a non-negligible set $B$ has the identity as an interior point of$\ B^{-1}B.$ 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 \textit{Haar} reference measure $\eta $. 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. This the first of four companion papers (we refer to the others as II [BinO11], III, [BinO12], and IV, [BinO13], below). Here (Propositions 1.1-1.7 and Theorems 11.-1.4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of \textit{selective} reference measures $\sigma $, drawing on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In II, we turn to a converse of the Steinhaus-Weil theorem, the Simmons-Mospan theorem, and related results. In III, we discuss Weil topologies, linking the topological group-theoretic and measure-theoretic aspects. We close in IV with some other interior-point results related to the Steinhaus-Weil theorem.