The Steinhaus-Weil property: I. Subcontinuity and amenability

摘要: The Steinhaus-Weil theorem that concerns us here is the simple, or classical, 'interior-points' property - in a Polish topological group non-negligible set B has identity as an interior point of Bˆ-1B: There are various converses; one mainly due to Simmons and Mospan. Here locally compact, so we have Haar reference measure η. Simmons-Mospan states (regular Borel) such if only it absolutely continuous with respect measure. This first four companion papers (we refer others II [BinO11], III, [BinO12], IV, [BinO13], below). (Propositions 1-7 Theorems 1-4) exploit connection between interior-points selective form infinitesimal invariance afforded by certain family measures σ, drawing on Soleckiis amenability at 1 (and using Fuller's notion subcontinuity). In II, turn converse theorem, Simmons- Mospan related results. discuss Weil topologies, linking group-theoretic measure-theoretic aspects. We close IV some other interior-point results Steinhaus- theorem.