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

TL;DR

This paper explores the Steinhaus-Weil property in Polish topological groups, examining its converse, subcontinuity, and amenability, and extends classical theorems with new insights into measure invariance and topological structures.

Contribution

It introduces new connections between interior-point properties, selective measures, and amenability, extending classical theorems to broader contexts and linking them with Weil-type topologies.

Findings

Characterization of measures with the Steinhaus-Weil property as absolutely continuous w.r.t. Haar measure

Development of relatives of the Simmons-Mospan theorem involving selective invariance

Linking interior-point properties with Weil-type topologies and Cameron-Martin space

Abstract

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.