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The Steinhaus-Weil theorem that concerns us here is the simple, or classical,`interior-points' property -- that in a Polish topological group anon-negligible set B has the identity as an interior point of $BB^{-1}$. Thereare various converses; the one that mainly concerns us is due to Simmons andMospan. Here the group is locally compact, so we have a Haar reference measure$\eta$. The Simmons-Mospan theorem states that a (regular Borel) measure hassuch a Steinhaus-Weil property if and only if it is absolutely continuous withrespect to the Haar measure. In Part I (Propositions 1-7, Theorems 1-4) weexploit the connection between the interior-points property and a selectiveform of infinitesimal invariance afforded by a certain family of selectivereference measures $\sigma$, drawing on Solecki's amenability at 1 (and usingFuller'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 IVwith Propositions 12-13 and Theorem B -- concerning the `compositeinterior-point' property (of $AB^{-1}$) and the Borell `relativeinterior-point' property (relative to the Cameron-Martin space) -- andcomplements.

The Steinhaus-Weil property: its converse, subcontinuity and Solecki amenability