The Steinhaus-Weil property: its converse, subcontinuity and Solecki 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 $BB^{-1}$. 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 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. In Part I (Propositions 1-7, 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 $\sigma$, drawing on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In Part II (Propositions 8, 9, Theorems 5, 6) we develop a number of relatives of the Simmons-Mospan theorem. In Part III (Theorems 7, 8) we link this with topologies of Weil type. We close in Part IV with Propositions 12-13 and Theorem B -- concerning the `composite interior-point' property (of $AB^{-1}$) and the Borell `relative interior-point' property (relative to the Cameron-Martin space) -- and complements.