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 Bˆ-1B: 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 reference measure η. 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 1-7 and Theorems 1-4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of selective reference measures σ, drawing on Soleckiís 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.