  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.
